METHOD FOR CALCULATING DURABILITY INDICATORS OF TURN INSULATION IN WINDINGS OF LOW-VOLTAGE ELECTRICAL MACHINES

A. P. Matyalis, Yu. P. Pokhodkov, E. K. Strelbitskii

(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)

One of the main tasks of the theory and practice of reliability of electrical machines is the creation of methods for calculating the probability of their failure-free operation. Correct mathematical models that reflect reality can be obtained only with due regard for the physics of the failure phenomenon. The quantitative correspondence of calculation results to actual results is determined by the initial assumptions, the variables taken into account, and their interrelations.

The accumulated experimental material shows that determining the pre-failure state of insulation from changes in insulation properties common to the entire volume is practically impossible. Resistance, the dielectric-loss tangent, and even breakdown voltage change only slightly, and then failure occurs. This leads to the idea that the cause of failure is a sharp change, in magnitude and in time, of properties within a limited volume.

A study of the influence of the technological process of winding manufacture on insulation properties [1, 2] and of the aging process, as well as a comparison of the strength of undamaged insulation with the level of stresses acting in service, made it possible to establish that the physical cause of breakdown of the insulation of low-voltage electrical machines is the presence of defects. A defect is understood to mean through damage to the insulation (technological damage, cracks, delaminations, etc.).

The previously developed method for calculating the durability of random windings of asynchronous motors has a drawback stemming from the concept that, upon failure, the insulation is punctured [3]. The method set forth below for calculating the probability of failure proceeds from the fact that, in service, failure occurs as a result of breakdown of air gaps at defective locations by the operating voltage or by switching overvoltages.

From the point of view of reliability, turn insulation is represented as a series connection of elements. An element is taken to be a defect on one of the contacting turns of the winding.

Failure of turn insulation occurs as a result of the appearance of a short circuit between adjacent turns. A short circuit arises as a result of breakdown of the insulation by the operating voltage or by switching overvoltages. In this connection it should be taken into account that not every breakdown (surface flashover over the insulation) by pulses of switching overvoltage leads to winding failure.

The conditional probability of the occurrence of a short circuit, if surface flashover of the insulation by a commutation overvoltage pulse has occurred, depends on the magnitude of the flashover distance \(z\), the magnitude of the overvoltage pulse \(v\), and the magnitude of the operating voltage \(U\). This dependence has been determined experimentally. The following empirical formula was obtained for determining the probability of a short circuit occurring during one flashover by a commutation overvoltage:

\[ P\{K|\Pi\}=1-\exp\left(-\frac{0.00701\cdot U+0.000154\,V}{z}\right). \tag{1} \]

If there are defects on adjacent turns and the distance between the copper of these turns at the defect locations is equal to \(z\), then the probability of breakdown of this gap by the operating voltage is

\[ q_z(U_c)=\int_0^{U_c} f(V/U_c)\left[\int_0^V f(U_z)\,dU\right]dV, \tag{2} \]

where
\(f(U_z)\) — probability density of the breakdown voltage of a gap of length \(z\);
\(U_c\) — amplitude of the maximum voltage value (allowing for the oscillation of the line voltage) applied to the section;
\(f(V/U_c)\) — conditional probability density of the voltage \(V\) between adjacent turns, if the voltage on the section is equal to \(U_c\).

The probability of breakdown of the gap \(z\) by one commutation overvoltage pulse and of the occurrence of a short circuit is

\[ q_z(V_c)=\int_0^\infty f(V_c)\left\{\int_0^{V_c} f(V/V_c)\left[\int_0^V f(U_z)\,dU\cdot P\{K|\Pi\}\right]\cdot dV\right\}\cdot dV_c, \tag{3} \]

where \(f(V_c)\) — probability density of the distribution of the magnitude of commutation overvoltages on the winding section.

The probability of breakdown of one element over the time interval \(\Delta \tau\), taking into account all possible distances \(z\), if the defectiveness remains constant, is

\[ q_{1\Delta\tau}=\int_0^\infty f(z)\{1-[1-q_z(V_c)]^{f_p\cdot\Delta\tau}[1-q_z(U_c)]\}\,dz, \tag{4} \]

where \(f(z)\) — probability density of the distribution of the magnitude of the gap between the copper at the defect location of one of the adjacent turns and the copper at the defect location of the second turn;
\(f_p\) — calculated number of commutation operations per unit time during which overvoltages arise that exceed the value \(U_c\).

Integral (4) is evaluated by a numerical method and can be used for any low-voltage electrical machines, if the distribution densities entering expressions (3) and (4) are known.

Defects in the insulation are distributed randomly. The distribution of their number is described by the Poisson law, and the magnitude of the gaps between defects has an exponential distribution with parameter \(\lambda\). The distance \(y\) from a fixed defect on one of the adjacent turns to a defect on the other turn, which may be located on either side of the fixed one, will have the density

\[ f(y)=2\lambda\cdot\exp(-2\lambda y);\quad y\geq 0, \tag{5} \]

where \(\lambda\) is the mean number of defects per unit length of wire.

The quantity \(z\) is equal to the sum of the distance \(y\) and the double-sided insulation thickness of the wire \(x\)

\[ z=x+y. \tag{6} \]

The insulation thickness is normally distributed with density

\[ f(x)=\frac{1}{\sqrt{2\pi}\sigma_x}\exp\left[-\frac{(x-\bar{x})^2}{2\sigma_x^2}\right], \tag{7} \]

where \(\bar{x}\) is the mean value of the double-sided thickness of the wire insulation;
\(\sigma_x\) is the root-mean-square deviation of the double-sided insulation thickness.

The distribution of the distance \(z\) is described by the composition of the distribution laws of the quantities \(x\) and \(y\)

\[ f(z)=\int_0^\infty 2\lambda e^{-2\lambda y}\cdot \frac{1}{\sqrt{2\pi}\sigma_x} \exp\left[-\frac{(z-y-\bar{x})^2}{2\sigma_x^2}\right]\,dy = \]

\[ =2\lambda\cdot e^{-2\lambda(z-\bar{x}-\lambda\sigma_x^2)} F\left(\frac{z-\bar{x}-2\lambda\sigma_x^2}{\sigma_x}\right), \tag{8} \]

where

\[ F\left(\frac{z-\bar{x}-2\lambda\sigma_x^2}{\sigma_x}\right) \]

is the normal distribution function.

The distribution of commutation overvoltages has now been obtained for asynchronous motors and, in the general case, is described by a superposition of distribution laws [3, 4]. If the values of overvoltage pulses equal and close to the rated voltage are neglected, then the remaining part of the distribution can be approximated by a normal law, and the value \(f_p\) should be taken equal to \((0.7 \div 0.8)i\), where \(i\) is the frequency of switching-on or reversing of motors.

In low-voltage asynchronous machines, overvoltages are distributed uniformly over the sections and turns of the winding [4]. The voltage between a pair of contacting turns with a difference in numbers, according to the winding diagram, equal to \(l\), will have the distribution density

\[ f(V/l)=\frac{1}{\sqrt{2\pi}\sigma_{Vx}} \exp\left[-\frac{(V-\kappa \bar{V}_c)^2}{2\sigma_{Vk}^2}\right], \tag{9} \]

where

\[ \bar{V}_c=\frac{\bar{V}_\phi}{n_c}; \]

\(\bar{V}_\phi\) is the mean value of the magnitude of the phase commutation overvoltages;
\(n_c\) is the number of sections in a phase;

\[ \kappa=\frac{l}{S}; \]

\(S\) is the number of turns in a section;

\[ \sigma_{Vk}=\kappa\sigma_{Vc}; \]

\[ \sigma_{Vc}=\frac{\sigma_{V\phi}}{n_c}; \]

\(\sigma_{V\phi}\) is the root-mean-square deviation of the magnitude of the phase commutation overvoltages.

For induction motors \(V_{\phi}=1.55\ \mathrm{kV}\) and \(\sigma_{V\phi}=0.54\ \mathrm{kV}\) [3]. It was established experimentally that the distribution of the quantity \(\kappa\) for pairs of touching turns of random windings has the density

\[ f(\kappa)=3(\kappa-1)^2,\quad 0\leqslant \kappa \leqslant 1, \tag{10} \]

which is also confirmed by the results given in [5]. Breakdown of the gap \(z\) occurs under the condition

\[ \Delta U=V-U_z>0. \]

The quantities \(U_z\) and \(V\) are normally distributed; therefore their difference also has a normal distribution with parameters

\[ \left\{ \begin{aligned} \sigma_{\Delta U\kappa}^{2} &= \sigma_{V\kappa}^{2}+\sigma_{Uz}^{2},\\ \overline{\Delta U}_{\kappa} &= \kappa \overline{U}_{c}-\overline{U}_{z} \end{aligned} \right. \tag{11} \]

and density

\[ f(\Delta U_{\kappa})= \frac{1}{\sqrt{2\pi}\,\sigma_{\Delta U\kappa}}\cdot \exp\left[-\frac{(\Delta U-\overline{\Delta U}_{\kappa})^2}{2\sigma_{\Delta U\kappa}^{2}}\right]. \tag{12} \]

The quantities \(\overline{U}_{z}\) and \(\sigma_{Uz}\) are determined experimentally by breaking down pairs of artificially damaged wire specimens at the corresponding values of \(z\) and under the ambient conditions corresponding to those in which the winding is in service.

Thus, approximating the distribution of switching overvoltages by a normal law leads to a substantial reduction in the volume of calculations. Expression (3) is reduced to the form

\[ q_z(V_c)=3\int_{0}^{1} F\left(\frac{\Delta \overline{U}_{\kappa}}{\sigma_{\Delta U\kappa}}\right) \left[ 1-\exp\left( -\frac{0.00701\,U_c+0.000154\,\overline{V}_{c}}{z}\,\kappa \right) \right]\times \]

\[ \times(\kappa-1)^2\,d\kappa. \tag{13} \]

To calculate the mean probability of breakdown of turn insulation, it is necessary to know the mean value of \(\lambda\). On the basis of a statistically planned experiment carried out to study defect formation in PETV wire insulation, the following dependence of the quantity \(\lambda\) on the acting loads was obtained:

\[ \lambda(\tau,\Theta,f)=\lambda_{0}+\tau c\cdot e^{\,b_{1}(\Theta-\Theta_{0})+b_{11}(\Theta-\Theta_{0})^{2}+b_{2}f}, \tag{14} \]

where \(\lambda_{0}\) is the number of defects per unit length of wire before aging \(\left(\frac{1}{\mathrm{mm}}\right)\);

\[ \tau \text{ is the aging time (hours);} \]

\[ \Theta \text{ is the operating temperature }({}^{\circ}\mathrm{C}); \]

\[ f \text{ is the reversing or switching frequency }(1/\text{hour}); \]

\[ \Theta_{0} \text{ is the temperature class of the insulation }({}^{\circ}\mathrm{C}); \]

\[ c=0.325\cdot 10^{-6};\quad b_{1}=0.631\cdot 10^{-1};\quad b_{11}=-0.39\cdot 10^{-3};\quad b_{2}=0.148\cdot 10^{-2}. \]

The principal influence on defect formation is exerted by the temperature of the winding. During operation of electrical machines it is a random variable distributed according to the normal law. The mean temperature is determined by the mean ambient temperature and the mean excess of the winding temperature over the ambient temperature. The variance is determined by the “skewness” of temperatures in the winding, changes in ambient temperature, fluctuations of the supply voltage, and changes in operating modes. The mean value of the number of defects per unit length at time \(\tau\) is determined as the mathematical expectation of the function [6]

\[ \overline{\lambda}_{\tau}=\int_{-\infty}^{\infty}\lambda(\Theta,\tau,f)f(\Theta)\,d\Theta, \tag{15} \]

where

\[ f(\Theta)=\frac{1}{\sqrt{2\pi}\cdot\sigma_{\Theta}}\cdot \exp\left[-\frac{(\Theta-\overline{\Theta})^{2}}{2\sigma_{\Theta}^{2}}\right]. \]

Substituting the values of the integrand functions into (15) and carrying out the transformations, we obtain

\[ \overline{\lambda}_{\tau}=\lambda_{0}+ \]

\[ +\tau c\,\frac{1}{\sqrt{1+2\sigma_{\Theta}^{2}b_{11}}}\, \exp\left[ \frac{ b_{1}(\overline{\Theta}-\Theta_{0}+0.5\,b_{1}\sigma_{\Theta}^{2}) +b_{11}(\overline{\Theta}-\Theta_{c})^{2} }{ 1+2\sigma_{\Theta}^{2}b_{11} } +b_{2}f \right]. \tag{16} \]

When calculating the probability of breakdown of an element, the upper limit of integral (4) may be taken as equal to \(z_m=0.5\text{--}1\) mm, since the probabilities of breakdown of larger gaps, as the calculations performed have shown, are negligibly small. Replacing integral (4) by a sum, we obtain the following formula for calculating the mean probability of breakdown of one element over the operating-time interval \(\tau_j-\Delta\tau,\ \tau_j\):

\[ \overline{q}_{1\Delta\tau j} = \frac{2\overline{\lambda}_{\tau j}z_m}{M_1} \sum_{i=0}^{i=M_1} e^{-2\overline{\lambda}_{\tau j}\left(\frac{z_m i}{M_1}-\overline{x}-\overline{\lambda}_{\tau j}\sigma_x^{2}\right)} F(i) \{1- \tag{17} \]

\[ -[1-q_V(i)]f_p\Delta\tau[1-q_U(i)]\}, \]

where

\[ F(i)=F\left( \frac{z_m i-M_1\overline{x}-2M_1\overline{\lambda}_{\tau j}\sigma_x^{2}} {M_1\sigma_x} \right); \tag{18} \]

\[ q_V(i)=\frac{3}{M_2}\sum_{i_1=0}^{i_1=M_2} F\left( \frac{i_1\overline{V}_c-M_2\overline{U}_z} {\sqrt{i_1^{2}\sigma_{V_c}^{2}+M_2^{2}\sigma_{U_z}^{2}}} \right) \left(\frac{i_1}{M_2}-1\right)^{2}; \qquad z=\frac{i\cdot z_m}{M_1}; \tag{19} \]

\[ q_U(i)=\frac{3}{M_3}\sum_{i_2=0}^{i_2=M_3} F\left( \frac{i_2 U_c-M_3\overline{U}_z} {M_3\sigma_{U_z}} \right) \left(\frac{i_2}{M_3}-1\right)^{2}; \qquad z=\frac{i\cdot z_m}{M_1}. \tag{20} \]

The value of \(\Delta\tau\) is chosen so that the values of the defectiveness level \(\overline{\lambda}_{\tau}\) at the times \(\tau-\Delta\tau\) and \(\tau\) differ insignificantly. The number of terms in the series is chosen from the calculation so that the ratio \(\frac{z_m}{M_1}\) is equal to \(0.02\div0.03\) mm. It is sufficient to take the numbers of terms \(M_2\) and \(M_3\) equal to \(10\div20\).

The conditional probability of failure of the machine due to turn insulation over the time interval \(0,\ \tau_j-\Delta\tau,\ \tau_j\), under the condition that no failure occurred over the time interval \(0,\ \tau_j-\Delta\tau\), is equal to

\[ q_{\Delta\tau j}=1-(1-\overline{q}_{1\Delta\tau j})^{N_e}, \tag{21} \]

where \(N_e=\psi L\,\overline{\lambda}_{\tau j}\) is the number of elements over the entire winding;

\(L\) is the total length of pairs of adjacent turns in the winding;

\(\psi\) is a coefficient accounting for the fraction of neighboring turns that are in close contact.

It is assumed that failure is possible only in the case of close contact between turns. This is due to the fact that, under normal operating conditions, the level of operating voltages is insufficient for breakdown of air gaps exceeding the insulation thickness, and after the action of short-time pulses of commutation overvoltages the strength of the air gap is restored. In the case of close contact there is no air gap between the insulated conductors, and, upon breakdown (surface flashover) by a sufficiently powerful commutation-overvoltage pulse, conducting bridges are formed on the surface of the insulation, leading to a short circuit of the turns under the operating voltage.

The probability of close contact is

\[ \psi = 0.93\sqrt{k_{\text{fill}}}, \tag{22} \]

where \(k_{\text{fill}}\) is the slot fill factor used in the practice of electromagnetic calculation.

The quantity \(L\) is determined by formula [7]

\[ L = (w_{\text{n}} + 1.5 w_{\text{v}} - 1.5)\, l_{w} z_{\text{p}}, \tag{23} \]

where \(w_{\text{n}}\) is the number of conductors in the outer layer of the section;
\(w_{\text{v}}\) is the number of conductors in the inner layers of the section;
\(l_{w}\) is the average turn length;
\(z_{\text{p}}\) is the number of slots in the machine.

It is convenient to carry out the calculation using equal time intervals \(\Delta \tau\). The probability of failure of turn insulation in the time interval \(\tau_j - \Delta \tau, \tau_j\) is

\[ Q_{\Delta \tau j} = q_{\Delta \tau j}\left(1 - Q_{\tau j-\Delta \tau}\right), \tag{24} \]

where \(Q_{\tau j-\Delta \tau}\) is the probability of failure of turn insulation up to the time instant \(\tau_j-\Delta \tau\).

The probability of failure over the time \(\tau = n\Delta \tau\) is

\[ Q_{\tau} = \sum_{j=1}^{n} Q_{\Delta \tau j}. \tag{25} \]

In conclusion, we note that, for asynchronous motors under normal operating conditions, the probability of breakdown of an element by the operating voltage, determined from formula (2), is considerably lower than the probability of breakdown by commutation overvoltages. Therefore, in practical calculations it may be neglected. This probability becomes significant when the strength of the element decreases, for example, during moisture condensation if the motor winding is in an atmosphere of high humidity.

REFERENCES

1. Yu. P. Muravlev, Yu. P. Pokholkov, E. K. Strelbitskii. The influence of winding-insulation work on the breakdown voltage of turn insulation in asynchronous motors. Izv. vuzov SSSR, “Elektromekhanika,” 1966, No. 1.

2. E. K. Strelbitskii, O. P. Muravlev, Yu. P. Pokholkov, A. S. Gitman. Investigation of defects in turn and frame insulation of random-wound windings of asynchronous motors. Izvestiya TPI, vol. 160, Tomsk, TSU Publishing House, 1966.

3. Method for calculating the durability of random windings of asynchronous electric motors with power from 0.18 to 100 kW. OTE, 139, 191, MEPI, SKBEM, Tomsk, 1969.

4. O. D. Goldbert. Theoretical and experimental development of methods for calculating reliability indices, accelerated testing, and quality control of asynchronous motors. Author’s abstract of dissertation. Moscow, 1971.

5. O. D. Goldbert. Quality and Reliability of Asynchronous Motors. Moscow, “Energiya,” 1968.

6. E. S. Venttsel. Probability Theory. Moscow, Fizmatgiz, 1962.

7. A. S. Gitman, O. P. Muravlev, Yu. P. Pokholkov, E. K. Strelbitskii. Method for calculating the reliability of turn insulation of asynchronous motors during the running-in period. Izvestiya TPI, vol. 190, Tomsk, TSU Publishing House, 1968.