Probabilistic Method for Calculating the Reliability of Insulation in Random-Wound Windings of Induction Motors
Abstract Generated abstract
The paper presents a probabilistic method for estimating the reliability of insulation in random-wound windings of general-purpose induction motors during the running-in period and early normal operation. It models turn, ground, and phase-to-phase insulation failures as functions of technological defects, experimentally determined breakdown-voltage distributions, winding geometry, impregnation and slot parameters, and switching overvoltage levels. The method uses Poisson and normal distribution assumptions to calculate elementary breakdown probabilities and combines them into overall winding-insulation reliability, while noting that aging effects require additional data on defect formation under operating stresses. Verification on 100 AO2-32-4 motors showed satisfactory agreement between calculated and experimental reliability values.
Full Text
PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV, HOLDER OF THE ORDER OF THE OCTOBER REVOLUTION
AND THE ORDER OF THE RED BANNER OF LABOR
Vol. 242
1972
PROBABILISTIC METHOD FOR CALCULATING THE RELIABILITY OF INSULATION IN RANDOM-WOUND WINDINGS OF INDUCTION MOTORS
Yu. P. POKHOLKOV, E. K. STREL’BITSKII
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
The reliability of electrical machines for general industrial use is determined mainly by the reliability of the insulation of their windings. The causes of insulation failures in electrical machines are currently being studied in the USSR and abroad [1, 2]. The principal cause of failures in low-voltage electrical machines is the presence of through defects in turn and ground insulation [3]. Through defects in insulation, consisting of punctures, cuts, and cracks, may arise both during the manufacture of the insulating materials and wires themselves and during the manufacture or operation of the machine.
The detection of defects during operation of the machine occurs either under pulses of switching overvoltages or under operating voltage, if the winding is heavily moistened or contaminated.
An assessment of the defectiveness of the insulation and of the distribution parameters of the breakdown voltage of defective spots, as well as an assessment of the level of switching overvoltages, makes it possible to calculate the reliability of winding insulation.
At the Tomsk Polytechnic Institute, a method has been developed for calculating the reliability of the insulation of random-wound windings of induction motors during the running-in period [4]. Experimental verification of the method and the principal work on introducing it into the practice of design bureaus and electrical-machine-building enterprises were carried out by engineers of the Special Design Bureau of the “Sibelektromotor” plant: L. I. Shtuden, V. M. Fedorov, and I. E. Myshkov.
At present, electrical-machine-building enterprises in the USSR use this method to assess the guaranteed reliability of winding insulation during the running-in period.
Calculation of the reliability of turn insulation in random-wound windings of induction motors
The defectiveness of wire insulation is determined from the results of testing the wire for breakdown by alternating voltage. Tests are performed on 150–250 wire specimens extracted from the slots of the machine after the windings have been laid (before impregnation). During the tests, insulation breakdown occurs between the metal of the wire and a flat metal electrode. The length of the wire specimen is 50–100 mm.
We define the insulation defectiveness \(q\) as the fraction of wire specimens having through defects
\[ q=\frac{n_{\overline{U}_1+3\sigma_1}}{\sum_{i=1}^{i} n_i}, \tag{1} \]
where
\[ n_{\overline{U}_1+3\sigma_1} \]
is the number of wire specimens punctured by a voltage not exceeding \(\overline{U}_1+3\sigma_1\);
\[ \overline{U}_1,\sigma_1 \]
are, respectively, the mean value and the standard deviation of the breakdown voltage of a wire having an evident through defect.
These parameters are determined in testing wires with artificially damaged insulation.
\[ \sum_{i=1}^{i} n_i \]
is the total number of tested specimens.
Let us proceed to the defectiveness of an elementary section of insulation of a single wire.
We shall define the radius of an elementary section of insulation \((\delta)\) on the basis of the maximum voltage at the defective spot and the electric strength of air at the defect site, taking into account surface flashover \((E_{\mathrm{B}})\),
\[ \delta=\frac{\overline{U}_1+3\sigma_1}{E_{\mathrm{B}}}. \tag{2} \]
We denote by \(n\) the number of elementary sections \((n)\) on the insulation area characterized by defectiveness \((q)\):
\[ n=\frac{d_{\mathrm{iz}}\cdot l_{\mathrm{obr}}\cdot k}{\delta^2}, \tag{3} \]
where
\[ d_{\mathrm{iz}} \]
is the diameter of the insulated wire;
\[ l_{\mathrm{obr}} \]
is the length of the wire specimen being tested;
\[ k \]
is a coefficient accounting for the area of the insulation being tested.
For wires of diameter \(d_{\mathrm{iz}}\leq 0.95\ \mathrm{mm}\), \(k=1\); for wires \(d_{\mathrm{iz}}>0.95\ \mathrm{mm}\),
\[ k=\frac{0.95}{d_{\mathrm{iz}}}. \]
Assuming that through damage to the insulation of a wire specimen is a rare event, we take the distribution of defects along the length of the wire to obey Poisson’s law [5]:
\[ 1-q=e^{-a}, \tag{4} \]
where
\[ a \]
is the mean number of insulation damages on the tested area of the wire specimen.
Then the probability \(P\) of damage to an elementary section of insulation of a single wire is determined as follows:
\[ P=\frac{a}{n}. \tag{5} \]
The probability of coincidence of defective elementary insulation sections in two conductors lying side by side, according to the multiplication theorem of probabilities, will be equal to \(P^2\). Breakdown between two closely touching conductors with coincident defects will occur with probability \((q')\)
\[ q' = p^2 F\left(\frac{U-\overline{U}_2}{\sigma_2}\right), \tag{6} \]
where \(F\left(\dfrac{U-\overline{U}_2}{\sigma_2}\right)\) is the normal distribution function of the breakdown voltage of the insulation of a pair of closely touching damaged conductors with coincident defects;
\(\overline{u}_2\) and \(\sigma_2\) are, respectively, the mean value and standard deviation of the breakdown voltage of a pair of damaged conductors with coincident defects;
\(\overline{u}_2\) and \(\sigma_2\) are determined experimentally on artificially damaged conductors.
Taking into account that in the machine slot there are distances between conductors distributed according to an exponential law, that the voltage along the phase is distributed linearly, and that the conductors are uniformly mixed, we write an expression for estimating the probability of breakdown of an elementary section of a pair of conductors \((q_{\text{el}})\) in the machine slot:
\[ q_{\text{el}} = p^2 \lambda \int_0^{U_k} e^{-\lambda U}\, F\left(\frac{U-U_2}{\sigma_2}\right) \left(1-\frac{U}{U_k}\right)\,dU, \tag{7} \]
where
\[ \lambda \]
is the parameter of the exponential law of distribution of the breakdown voltage between conductors in the machine slot.
The parameter \(\lambda\) is determined from the slot fill factor \((k_z)\), the mean electric strength of the impregnating compound \((\overline{E}_p)\), and the mean fraction of the volume of impregnating compound in the gap between conductors \(v_p\):
\[ \lambda = \frac{1}{ \left(1.075\sqrt{\frac{1}{k_z}-d_{\text{unins}}}\right)\cdot \overline{E}_p\cdot v_p}, \tag{7a} \]
\(d_{\text{unins}}\) is the diameter of the uninsulated wire;
\(u_k\) is the voltage, taking into account commutation overvoltages, falling on an individual insulated coil of the phase.
The level of the amplitudes of the pulses of commutation overvoltages applied to the winding with a probability close to 1 increases with an increase in the number of switchings \(\nu\) and, consequently, in the operating time of the machine \(\tau\).
Note. Under normal operating conditions of the machine, the time in this calculation may be taken as no more than 5000 hours.
The dependence of the amplitudes of commutation overvoltages on the phase of the motor winding \(U_\phi\) on the number of machine switchings \(\nu\) may be estimated by the formula obtained as a result of processing experimental data:
\[ e^{2.25U_\phi-0.7} = \nu . \tag{8} \]
The quantity \(U_k\) depends on the number of separately insulated coils in the phase \(n_k\) and, consequently, on the type of machine winding:
\[ U_k = \frac{U_\phi}{n_k}. \tag{9} \]
In this connection, expression (7) makes it possible to estimate the probability of breakdown of an elementary section of insulation over a specified interval of time—
This latter circumstance makes it possible to calculate also the probability of failure-free operation of the turn insulation of the winding over a specified time, i.e., the reliability \(R_{\text{turn}}\)
\[ R_{\text{turn}}=(1-q_{\text{el}})^N, \tag{10} \]
where
\(N\) is the number of elementary sections of turn insulation in the entire machine winding.
\[ N=n\,(S_{\text{out}}+1.5S_{\text{in}}-1.5)\frac{\overline{l_w}}{l_{\text{turn}}}\cdot Z, \tag{11} \]
\(S_{\text{out}}\) and \(S_{\text{in}}\) are, respectively, the number of conductors in the outer and inner layers of the winding in the slot;
\(\overline{l_w}\) is the average length of a winding turn;
\(Z\) is the number of stator slots.
Calculation of the Reliability of the Ground Insulation of Induction-Motor Windings
In calculating the reliability of the ground insulation of windings, the insulation is represented as a series of consecutively connected capacitors. The initial data for the calculation are the distributions of breakdown voltages of the individual layers of insulating materials, obtained experimentally after the winding has been laid (before impregnation), and the level of switching overvoltages. Samples of ground insulation carefully extracted from stator slots are subjected to testing. The tests are carried out with alternating voltage in “needle–plane” type electrodes. The area of the elementary insulation section for which the breakdown probability is first determined is taken to be the area of the flat electrode used in the tests (\(78.5\ \text{mm}^2\)). The breakdown probability of an elementary section of two-layer insulation \(P(AB)\), with layers \(A\) and \(B\), is determined as follows:
\[ P(AB)=P(A)P_A(B)+P(B)P_B(A)-P(A)P(B), \tag{12} \]
where
\(P(A)\,[P(B)]\) are, respectively, the probabilities of breakdown of layer \(A\), \(B\) when layer \(B\), \(A\) has not broken down;
\(P_B(A)\,P_A(B)\) are, respectively, the probabilities of breakdown of layers \(A\), \(B\) when layers \(B\), \(A\) have broken down.
The breakdown probability of an elementary section of three-layer insulation \(P(ABC)\)
\[ \begin{aligned} P(ABC)=&\,P(AB)\cdot P_{AB}(C)+P(AC)\cdot P_{AC}(B)+{}\\ &+P(BC)\cdot P_{BC}(A)-P(A)\cdot P_A(B)\cdot P_A(C)-{}\\ &-P(B)P_B(A)P_B(C)-P(C)P_C(A)P_C(B)+P(A)P(B)P(C). \end{aligned} \tag{13} \]
The meanings of the symbols correspond to those in formula (12).
Note. The breakdown probability of four- and five-layer insulation compositions is determined by formulas analogous to formulas (12), (13).
The breakdown probabilities of the individual layers of insulating materials for various combinations of broken-down and unbroken layers in the composition are determined from the experimental distributions of breakdown voltage, taking into account the occurrence of a certain amplitude of switching overvoltage and the distribution of voltage among the insulation layers in accordance with their dielectric permittivities and thicknesses.
Thus, for example, the breakdown probability of an elementary section of any \(i\)-th insulation layer, when the other layers of a composition of \(n\) layers have not broken down, is determined at voltage \(U(i)\)
\[ U(i)= \frac{ U_{\phi}\dfrac{d_i}{\varepsilon_i}\prod_{i=1}^{n}\varepsilon_i }{ \sum_{i=1}^{n}\dfrac{d_i}{\varepsilon_i}\prod_{i=1}^{n}\varepsilon_i }, \tag{14} \]
where
\(U_{\phi}\) is the amplitude of the commutation overvoltage arriving at the winding phase for a specified number of switchings or operating time of the machine (see formula (8));
\(d_i\) is the thickness of the \(i\)-th insulation layer;
\(\varepsilon_i\) is the dielectric permittivity of the \(i\)-th insulation layer.
When calculating the voltage falling on an individual insulation layer when one or several layers have been punctured, \(\varepsilon\) of the punctured layers is set equal to unity, and the thickness to zero. The reliability of the ground insulation of the motor winding \((R_{\text{corp}})\)
\[ R_{\text{corp}}=\left[1-P_{\text{el}}\right]^{\frac{S_{\text{corp}}}{k'S_{\text{el}}}}, \tag{15} \]
where
\[ P_{\text{el}}=P(AB)\ \text{or}\ P(ABC)\ \text{or}\ P(ABCD)\ \text{etc.} \]
depending on the number of layers of which the ground insulation consists;
\(S_{\text{corp}}\) is the total area of the ground insulation of the machine winding;
\(S_{\text{el}}\) is the area of an elementary section;
\(k'\) is a coefficient taking into account the fraction of the ground-insulation area that is under a voltage sufficient for detecting defects.
The reliability of the phase-to-phase insulation \((R_{\text{mf}})\) is calculated analogously to the calculation of the reliability of the ground insulation. In this case, the reliability of the entire insulation of the random winding of an induction motor is determined as follows:
\[ R_{\text{ins. wind.}}=R_{\text{turn}}\cdot R_{\text{corp}}\cdot R_{\text{mf}}. \tag{16} \]
This reliability calculation takes into account only technological defects and is valid for such an operating time of the machine during which no intensive thermal aging of the insulation occurs that would lead to the appearance of new defects.
Data from experiments on the aging of the insulation of low-voltage machines under the influence of operating factors indicate that this time is about 5 thousand hours.
An experimental verification of the calculation results according to the method, carried out on 100 motors of type AO2-32-4, showed satisfactory agreement between the calculated and experimental data.
Table 1 gives the results of the calculations and the experiment.
Table 1
| Reliability of type of insulation | Data: experiment | Data: calculation |
|---|---|---|
| Turn-to-turn | 0.93 | 0.921 |
| Ground | 0.99 | 0.985 |
| Phase-to-phase | 0.98 | 0.987 |
| Winding insulation | 0.902 | 0.895 |
Note. The calculation was carried out for a number of switchings equal to 75 thousand. The experiment lasted 3 thousand hours at operating temperature and a reversing frequency of 25 reversals per hour.
The calculation of winding-insulation reliability with allowance for aging can be performed by the proposed method, provided that the dependence of insulation defectiveness on the level of the acting factors and on the duration of their action is known.
Conclusions
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A method is proposed for calculating the insulation reliability of random-wound windings of induction motors for general industrial use during the running-in period and normal operation, taking into account technological, design, and operational factors.
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To use the method for calculating the reliability of the insulation of machine windings over the period during which intensive aging of the insulation occurs, it is necessary to investigate the operating conditions of the insulation in windings and the rate of formation of defects in the insulation under the influence of these conditions.
References
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N. A. Kozyrev. Insulation of Electrical Machines and Methods for Its Testing. “Energiya,” 1963.
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I. S. Johnson. Processes of breakdown of insulation electromotors. “Electrical insulation materials and application,” 5-th conf. Chicago Ill. 1963.
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E. K. Strelbitskii, O. P. Muravlev, Yu. P. Pokholkov. The influence of winding-insulating operations on the breakdown voltage of turn-to-turn and ground insulation of induction motors. Izv. vuzov, “Elektromekhanika,” 1966, No. 1.
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E. K. Strelbitskii, Yu. P. Pokholkov, O. P. Muravlev, L. I. Shtuden, V. M. Fedorov. Method for Calculating the Reliability of the Insulation of Windings of Induction Electric Motors with Power from 0.18 to 100 kW. Tomsk Interbranch Territorial Center of NII, Tomsk, 1969.
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A. K. Mitropol’skii. Technique of Statistical Computations. Fizmatgiz, 1964.