Reliability Model of Turn-to-Turn Insulation in Random-Wound Windings During the Burn-In Period
Abstract Generated abstract
This paper develops a probabilistic reliability model for turn-to-turn insulation failures in random-wound electrical machine windings during the burn-in period. The winding is divided into elementary insulation gaps, classified by enamel damage and conductor contact, with defect probabilities estimated from conductor tests and contact probabilities related to slot fill using a queueing-theory analogy. The authors derive distributions for enamel and gap breakdown voltages, combine them with switching-overvoltage distributions to estimate the probability of breakdown for different elementary-section types, and show that failures are dominated by sections with both enamel layers damaged. The resulting model treats the number of defects per machine as Poisson distributed and gives formulas for failure probability and failure rate as functions of repeated overvoltage pulses during run-in.
Full Text
RELIABILITY MODEL OF TURN-TO-TURN INSULATION OF RANDOM-WOUND WINDINGS DURING THE RUN-IN PERIOD
A. P. MATYALIS, E. K. STREL'BITSKII
(Presented by the scientific seminar of the Department of Electrical Machines and by the Electrical Machines Division of the Scientific Research Institute of AEM)
An analysis of the quality of turn-to-turn insulation carried out by the Tomsk Polytechnic Institute showed the presence of a large number of weak points in conductors having an inadmissibly low breakdown voltage.
We shall agree to call a failure of turn-to-turn insulation at least one electrical breakdown between adjacent turns of a machine winding, i.e., a turn-to-turn short circuit in an operating or transient mode.
A failure occurs if, in at least one place, the voltage between adjacent turns exceeds the breakdown voltage of the insulating gap. The main cause of failure is the presence of conductors with damaged insulation, which we shall call defective conductors.
The machine winding is represented as consisting of a set of elements. An element is the insulating gap between adjacent conductors of length \(l_{\mathrm{el}}\) in the direction of the conductor axis. In the general case, an elementary section has a breakdown voltage
\[ U_{\mathrm{el}} = U_{\mathrm{e}} + U_{\mathrm{l}}, \tag{1} \]
where \(U_{\mathrm{e}}\) is the breakdown voltage of the enamel layers of adjacent conductors;
\(U_{\mathrm{l}}\) is the breakdown voltage of the gap between insulated conductors, filled with air or impregnating compound.
The purpose of dividing the winding into elementary sections is to obtain such a section length \(l_{\mathrm{el}}\) within whose limits the breakdown voltage may be considered unchanged. For the currently existing thicknesses and breakdown voltages of the enamel coating of round-section conductors, it may be assumed that \(l_{\mathrm{el}} = 1\) mm.
We shall classify elementary sections according to the presence or absence of individual layers in their composition.
Let us denote:
\(\bar{A}\) — the event consisting in the fact that one enamel layer of an elementary section is damaged. It will be shown below that damage is equivalent to the absence of a layer. The probability of this event is \(P\{\bar{A}\} = p\).
\(\bar{A}\) — the event consisting in the fact that between the insulated conductors there is a gap filled with air or impregnating varnish. The probability of this event is \(P\{B\} = 1 - g\).
\(\bar{B}\) — the event consisting in the fact that the insulated conductors touch. The probability of this event is \(P\{\bar{B}\} = g\).
The complete event space is: $\bar A\bar A\bar B$, $\bar A A\bar B$, $A\bar A\bar B$, $(AA\bar B)$, $\bar A\bar A B$, $\bar A A B$, $A\bar A B$, $(AAB)$. Their probabilities are given in the table.
| Type of elementary section | Characteristic of the elementary section | $\bar B$ a |
$B$ b |
|---|---|---|---|
| 1 | $AA$ | $(1-p)^2 g$ | $(1-p)^2(1-g)$ |
| 2 | $\bar A A$ or $A\bar A$ | $2p(1-p)g$ | $2p(1-p)(1-g)$ |
| 3 | $\bar A\bar A$ | $p^2 g$ | $p^2(1-g)$ |
The total number of elementary sections of turn-to-turn insulation is calculated taking into account that each conductor theoretically may be in contact with six neighboring conductors if it is located in the middle of the slot, or with four neighboring conductors if it adjoins the slot ground insulation.
\[ N = (2w_{\mathrm{out}} + 3w_{\mathrm{in}} - 3)\frac{l_w}{2l_{\mathrm{el}}}\cdot z, \tag{2} \]
where
$w_{\mathrm{out}}$ is the number of conductors in the outer layer of the section;
$w_{\mathrm{in}}$ is the number of conductors in the inner layers of the section;
$l_w$ is the average length of a turn;
$z$ is the number of slots in the machine.
The probability of damage to an elementary section is determined from tests of conductors removed from an unwound, unimpregnated machine. The breakdown voltage of a damaged section is the breakdown voltage of an air layer of the corresponding thickness (more precisely, the flashover voltage along the surface of the corresponding length) and, consequently, differs significantly from the breakdown voltage of the enamel layer. This circumstance makes it possible to dispense with the laborious acquisition of histograms of breakdown voltages and to be satisfied with a binary classification of the insulation: damaged or undamaged. In this case the tests are carried out at a fixed test voltage, which considerably simplifies the testing procedure and increases labor productivity. The level of the test voltage and the size of the electrodes must be chosen so that, on the one hand, most of the damaged conductors break down, and, on the other hand, the probability of breakdown of undamaged sections is small. It must be borne in mind that the breakdown voltage increases as the conductor diameter increases if the tests are carried out on a flat electrode, and as the shot diameter increases if the tests are carried out in shot. The reason for the increase in breakdown voltage is the increase in the flashover length along the surface. With both test methods, practically all damaged sections are detected, but the parameters of the breakdown voltage may differ for the reasons indicated above.
Tests of conductors on a flat electrode rather well simulate the conditions of breakdown to a neighboring conductor. However, in most cases this involves flashover along the enamel surface, which leads to high breakdown voltages comparable with the breakdown voltages of undamaged enamel layers. This method imposes high requirements on the accuracy of measurement and on maintaining stable experimental conditions. The method of testing in shot is much less sensitive to measurement accuracy, since the counter-voltages are weak-
…locations is substantially less than the breakdown voltages of the enamel, and recognition of damaged and undamaged sections presents no difficulty.
The defectiveness value obtained in tests in bulk must be reduced by a factor of 5–6, since in the subsequent calculation the defectiveness is referred to one neighboring conductor.
We established the distribution law for distances between conductors and calculated the probability of close contact by methods of queueing theory. The analogue of the probability of zero waiting time when serving a customer in a queue is the probability of close contact, and the idle time of the device is analogous to the distance between conductors.
The probability of close contact is
\[ g=\Psi, \tag{3} \]
where \(\Psi\) is the linear slot-fill factor.
\[ \Psi=0.93\sqrt{K_{\text{fill}}}, \tag{4} \]
\(K_{\text{fill}}\) is the fill factor of the free slot area, used in practice in electromagnetic calculation.
The distribution law for distances between insulated conductors is
\[ Y(x)=1-(1-\Psi)e^{-\lambda x}, \tag{5} \]
where \(\lambda\) is the number of conductors per unit linear dimension of the slot (height, width).
Breakdown of an elementary section occurs under the action of a switching overvoltage \(V\), whose magnitude depends both on the difference between the numbers of neighboring turns according to the winding diagram and on the magnitude of the overvoltage impulse of the winding phase. We must determine the probability of the event
\[ P\{V>U_{\text{el}}\}, \tag{6} \]
where
\[ V=\frac{\Delta w}{w}V_{\phi}; \]
\[ U_{\text{el}}=U_{e1}+U_{e2}+U_l; \]
\(\Delta w\) is the difference between the numbers of neighboring turns according to the winding diagram;
\(w\) is the number of turns of a phase;
\(V\) is the magnitude of the switching-overvoltage impulse of the phase;
\(U_{e1}\) is the breakdown voltage of one enamel layer;
\(U_{e2}\) is the breakdown voltage of the second enamel layer;
\(U_l\) is the breakdown voltage of the gap between insulated conductors.
This problem is rather complex, since the quantities \(V_{\phi}\), \(U_e\), \(\Delta w\), and \(U_l\) are random variables distributed according to different laws.
The cases and types of elementary section.
Let us augment the space of events listed in the table by the events \(P\) and \(\overline{P}\), denoting, respectively, breakdown or nonbreakdown of an elementary section. The event \(\overline{A}A\overline{B}P\), for example, consists in the fact that an elementary section consisting of one undamaged enamel layer and one damaged enamel layer, in close contact with each other, is broken down by a switching overvoltage. We have to calculate the following probabilities:
\(P\{\overline{A}A\overline{B}P\}\) — the probability of breakdown of sections of type 1a,
\(P\{\overline{A}ABP\}\) — the probability of breakdown of sections of type 1b,
\(P\{AABP\}\)—the probability of breakdown of sections of type 2a,
\(P\{AABP\}\)—the probability of breakdown of sections of type 2b,
\(P\{AABP\}\)—the probability of breakdown of sections of type 3a,
\(P\{AABP\}\)—the probability of breakdown of sections of type 3b.
To carry out the stated task, it is necessary to determine the distribution laws of the random variables listed above, or their numerical characteristics.
We shall now determine the distribution laws of the breakdown voltages of the various types of elementary sections.
The technological process of applying enamel insulation consists in repeatedly passing the wire through dies of increasing diameter. Each new layer is positioned eccentrically with respect to the center of the wire; however, there exists a minimum one-sided layer thickness, caused by the wetting property of the varnish.
To establish the minimum one-sided thickness of the enamel coating, the breakdown voltages of the enamel coating were investigated after 1, 2, 3, 4, and 5 passes through the die.
The investigations showed that the distribution of the breakdown voltage of a single layer is not uniform, as might have been assumed in the absence of wetting. Estimation of the minimum thickness of one layer gives a value of \(1.2 \ldots 1.5\ \mu\), which corresponds to a minimum breakdown voltage
\[ U'_{\min}=200 \ldots 250\ \text{V}. \]
As the number of layers increases, the distribution of the sum of the breakdown voltages of the layers asymptotically tends to a truncated normal distribution with parameters
\[ \overline{U}=rU'_{\min}+(\overline{U}_i-U'_{\min})r=r\overline{U}_i; \]
\[ \sigma_U^2=\frac{r}{3}(\overline{U}_i-U_{\min})^2; \tag{7} \]
\[ U_{\min}=rU'_{\min} \]
and density
\[ \begin{cases} f(U)=C_{\mathrm{n}}\dfrac{1}{\sqrt{2\pi}\,\sigma_U} \exp\left[-\dfrac{(U-\overline{U})^2}{2\sigma_U^2}\right]; & U \geq U_{\min},\\[1.2em] f(U)=0; & U<U_{\min}, \end{cases} \tag{8} \]
where
\(C_{\mathrm{n}}\)—normalizing factor;
\(r\)—number of passes during enameling of the wire. Usually \(r=6\text{–}7\);
\(\overline{U}_i\)—mean breakdown voltage of the \(i\)-th layer.
When wire is tested in shot and in twists, breakdown occurs at the weakest point, i.e., at the point having the minimum thickness of the enamel layer. Therefore, the breakdown-voltage distribution of undamaged wire is the distribution of extreme values of a truncated normal distribution with density
\[ \begin{cases} f_{\mathrm{np}}(U_{\mathrm{p}})=C_{\mathrm{n}}^{1} n f(U) \left[1-\displaystyle\int_{U_{\min}}^{U} f(U)\,dU\right]^{n-1}; & U \geq U_{\min};\\[1.2em] f_{\mathrm{np}}(U_{\mathrm{p}})=0; & U<U_{\min},\\[1.2em] C_{\mathrm{n}}^{1}=\dfrac{1}{1-C_{\mathrm{n}}^{-n}}. \end{cases} \tag{9} \]
Here \(n\) is the number of segments on the length of wire within which the breakdown voltage may be regarded as unchanged.
Formula (9) can be used to determine the number of segments \(n\) whose breakdown voltage is not mutually correlated, and the length of an elementary segment \(l_{\mathrm{el}}\). It has thus been established that \(l_{\mathrm{el}} \approx 1\) mm.
The probability of breakdown of type 1a segments under the action of switching overvoltages can be calculated as follows.
The probability density of exceedance by a switching overvoltage of the breakdown voltage of the enamel layers is the density of the composition
\[ \Delta U = V - U_{\mathrm{e}} . \tag{10} \]
Since \(V\) and \(U_{\mathrm{e}}\) are normally distributed, their difference is also normally distributed with parameters
\[ \Delta \overline{U} = \overline{V} - \overline{U}_{\mathrm{e}}; \]
\[ \sigma_{\Delta U}^{2} = \sigma_{V}^{2} + \sigma_{U}^{2}. \tag{11} \]
The probability that \(\Delta U > 0\) is equal to
\[ P'\{\Delta U > 0\} = \int_{0}^{\infty} \frac{1}{\sigma_{\Delta U}\sqrt{2\pi}} \exp\left[-\frac{(U-\Delta \overline{U})^{2}}{2\sigma_{\Delta U}^{2}}\right]\,dU = F\left(\frac{\Delta \overline{U}}{\sigma_{\Delta U}}\right). \tag{12} \]
Breakdown of the enamel layer occurs when the magnitude of the overvoltage impulse exceeds the breakdown voltage of the layer; however, in this case the magnitude of the overvoltage impulse must exceed the minimum breakdown voltage of the layer.
Thus, the probability of failure of an elementary type 1a segment under a single action of a switching-overvoltage impulse is equal to
\[ q_{1} = P'\{\Delta U > 0\}\cdot P''\{V > U_{\min}\}, \tag{13} \]
where
\[ P''\{V > U_{\min}\} = F\left(\frac{\overline{V}-U_{\min}}{\sigma_{V}}\right) \tag{14} \]
is the probability that the overvoltage impulse exceeds the minimum breakdown voltage of the enamel layer.
The average probability of breakdown of an elementary segment is calculated with allowance for the distribution of the overvoltage magnitude along the coil:
\[ \overline{q}_{1} = 3\int_{0}^{1} F\left( \frac{\kappa \overline{V}_{k}-\overline{U}_{\mathrm{e}}} {\sqrt{\kappa^{2}\sigma_{V_k}^{2}+\sigma_{\mathrm{e}}^{2}}} \right) F\left( \frac{\kappa \overline{V}_{k}-U_{\min}} {\kappa\sigma_{V_k}} \right) (1-\kappa)^{2}\,d\kappa . \tag{15} \]
The probability of breakdown of at least one elementary type 1a segment in the machine under a single overvoltage impulse is equal to
\[ q_{1} = 1-(1-\overline{q}_{1})^{N_{1a}}, \tag{16} \]
where
\[ N_{1a}=(1-p)^{2}g\cdot N . \]
This probability may be neglected because of its smallness in comparison with the probability of breakdown of type 3 elementary segments.
The probability of breakdown of type 3 segments under the action of switching overvoltages is calculated by analogous formulas. First we calculate the parameters of the breakdown voltage of the air and varnish gaps. The thicknesses of the air and varnish gaps ...
are equal to the thicknesses of the double enamel layer and, consequently, are distributed according to a truncated normal law with parameters
\[ \begin{gathered} \bar{\delta}=\bar{\delta}_{\mathrm{e}},\\ \sigma=\sigma_{\mathrm{e}},\\ \delta_{\min}=2r\delta'_{\min}. \end{gathered} \tag{17} \]
The value \(\delta\) may be taken from a handbook, while the standard deviation of the double thickness is calculated by the formula
\[ \sigma \simeq \frac{2.4\bar{\delta}}{\sqrt{6r}}. \tag{18} \]
The breakdown voltage of air in thin layers, according to Paschen’s law, depends linearly on thickness, except for the initial interval \(\delta=0\)—\(0.008\ \mathrm{mm}\) at atmospheric pressure,
\[ U(\delta)=300+7000\delta, \tag{19} \]
where \(\delta\) is in mm.
The minimum value of the breakdown voltage of the air gap is \(U_{\min}=327\) volts.
Owing to linearity, \(U_{\mathrm{e}}\) may be regarded as normally distributed with parameters
\[ \begin{gathered} \bar{U}_{\mathrm{e}}=2E_{\mathrm{v}}\bar{\delta},\\ \sigma_{\mathrm{e}}=1.2\,\frac{\bar{U}_{\mathrm{e}}}{\sqrt{6r}} \end{gathered} \tag{20} \]
and density
\[ \left\{ \begin{aligned} f(U)&=\frac{1}{\sqrt{2\pi}\sigma_{\mathrm{e}}} \exp\left[-\frac{(U-\bar{U}_{\mathrm{e}})^2}{2\sigma_{\mathrm{e}}^2}\right];\quad U\geqslant 327\ \mathrm{V};\\ f(U)&=0;\quad U<327\ \mathrm{V}. \end{aligned} \right. \tag{21} \]
The probability that an overvoltage impulse exceeds the breakdown voltage of an elementary section of type 3 is
\[ P\{\Delta U>0\} = \int_{0}^{\infty} \frac{1}{\sqrt{2\pi}\sigma_{\Delta U}} \exp\left[-\frac{(U-\Delta\bar{U})^2}{2\sigma_{\Delta U}^2}\right]\,dU = F\left(\frac{\Delta\bar{U}}{\sigma_{\Delta U}}\right), \tag{22} \]
where
\[ \begin{gathered} \Delta\bar{U}=\bar{V}-\bar{U}_{\mathrm{e}},\\ \sigma_{\Delta U}^{2}=\sigma_{V}^{2}+\sigma_{\mathrm{e}}^{2}. \end{gathered} \tag{23} \]
The probability of breakdown of an elementary section of type 3 under the action of a single overvoltage impulse is
\[ q_{3}=F\left(\frac{\Delta U}{\sigma_{\Delta U}}\right)\cdot F\left(\frac{\bar{V}-U_{\min}}{\sigma_{V}}\right), \tag{24} \]
where \(F\left(\dfrac{\bar{V}-U_{\min}}{\sigma_{V}}\right)\) is the probability that the overvoltage impulse exceeds the minimum breakdown voltage of the air gap.
The mean probability of breakdown of an elementary section is calculated by a formula analogous to (15),
\[ \bar{q}_{3} = 3\int_{0}^{1} F\left( \frac{\kappa\bar{V}_{k}-U_{\mathrm{e}}} {\sqrt{\kappa^{2}\sigma_{Vk}^{2}+\sigma_{\mathrm{e}}^{2}}} \right) \cdot F\left( \frac{\kappa\bar{V}_{k}-U_{\min}} {\kappa\sigma_{Vk}} \right) (1-\kappa)^{2}\,d\kappa. \tag{25} \]
The integral (25) is not tabulated and for practical calculations may be replaced by the sum
\[ q'_3 \simeq \frac{3}{M}\sum_{i=M}^{i=0} F_1(i)F_2(i)\left(1-\frac{i}{M}\right)^2, \tag{26} \]
where
\[ F_1(i)=F\left(\frac{i\bar V_k-MU_\mathrm{e}}{\sqrt{i^2\sigma_{Vk}^2+M^2\sigma_\mathrm{e}^2}}\right); \tag{27} \]
\[ F_2(i)=F\left(\frac{i\bar V_k-MU_{\min}}{i\sigma_{vk}}\right). \tag{28} \]
It is more convenient to carry out the summation from \(i=M\) to \(i=0\). It is sufficient to take the total number of terms of the series as \(M=10\ldots 20\); moreover, the summation may already be stopped halfway, since \(F_2(i)\) decreases rapidly as \(i\) decreases. In practice, in operation, about \(20\%\) of weak points are detected.
Let us determine the failure rate during the run-in period. The mean number of defects in a winding is
\[ a=N z_a=Ngp^2. \tag{29} \]
This number is small: for induction motors of sizes 2–4,
\[ a=0.2\ldots 2.0. \]
Let \(x\) denote the number of defects in a given machine. The distribution of the number of defects among machines obeys the Poisson law
\[ P(x)=\frac{a^x}{x!}e^{-a}. \tag{30} \]
The fraction of machines having no defects is
\[ P(0)=e^{-a}. \tag{31} \]
The turn-to-turn insulation of these machines cannot fail during the run-in period.
The failure rate is the sum of the failure rates of machines having 1, 2, 3, etc. defects. The weighting coefficients of the sum are determined by formula (30).
The probability of failure of a machine having one defect under a single overvoltage pulse is
\[ Q_{\nu=1,\,x=1}=q'_{3a}. \tag{32} \]
The probability of failure of a machine under \(\nu\) overvoltage pulses is
\[ Q_{\nu,x}=1-(1-q'_{3a})^\nu. \tag{33} \]
The probability of failure of a machine having \(x\) defects under \(\nu\) overvoltage pulses is
\[ Q_{\nu,x}=1-(1-q'_{3a})^{x\nu}. \tag{34} \]
The failure rate as a function of \(\nu\) is
\[ \lambda(\nu)=e^{-a}\sum_{x=1}^{\infty}\frac{a^x}{x!}\frac{\partial}{\partial \nu}\left[1-(1-q'_{3a})^{x\nu}\right]. \tag{35} \]
For practical calculations one may use an approximate formula for the failure rate per one switching-on in the interval from \(\nu\) to \(\nu+\Delta\nu\).