Mathematical Model of the Reliability of Turn Insulation in Random-Wound Windings of Electrical Machines during the Running-In Period
Abstract Generated abstract
This paper develops a probabilistic model for estimating the reliability of turn insulation in random-wound windings of electrical machines during the running-in period. The model treats insulation gaps between adjacent conductors as elementary reliability elements, combines statistical distributions for enamel breakdown voltage and varnish or air gap breakdown voltage, and derives formulas for short-circuit probability, factory test detection, and early-service failures. Application to an A032-4A asynchronous motor suggests that many enamel defects are introduced during insertion into slots, that plant testing detects only a small fraction of weak points, and that reducing technological damage to wire insulation can substantially improve winding reliability.
Full Text
MATHEMATICAL MODEL OF THE RELIABILITY OF TURN INSULATION IN RANDOM-WOUND WINDINGS OF ELECTRIC MACHINES DURING THE RUNNING-IN PERIOD
A. S. Gitman, Yu. P. Lokhodkov, E. K. Strelbitskii
(Presented by the scientific seminar of the departments of electric machines and general electrical engineering)
The aim of the present work is to construct a theoretical-probabilistic model of the reliability of the turn insulation of electric machines, on the basis of which, from data on the quality of the wire and its damageability in the technological process, it would be possible to estimate the reliability of the machine in the most important interval—the running-in period.
The element of the model is the insulation gap between adjacent conductors, with breakdown voltage
\[ u_{\mathrm{el}} = u_{\mathrm{e}} + u_{\mathrm{l}}, \tag{1} \]
where
\(u_{\mathrm{e}}\) is the sum of the breakdown voltages of the enamel layers of adjacent wires,
\(u_{\mathrm{l}}\) is the breakdown voltage of the gap located between the insulated wires.
Since along the length of the wire the quantities \(u_{\mathrm{e}}\) and \(u_{\mathrm{l}}\) vary, we divide the wire into sections of length \(\Delta l\), over which the indicated quantities remain unchanged. Thus, a winding of total length \(L\) consists of \(N' = \dfrac{L}{\Delta l}\) elementary segments, or of \(N'' = \dfrac{L}{l_{\mathrm{st}}}\) standard segments. The standard length of a specimen in tests of electric strength is \(l_{\mathrm{st}} = 125\) mm. The total number of elementary gaps is
\[ N_1 = \nu N'. \]
\(\nu\) depends on the configuration of the section cross-section in the slot and in the end parts and lies within the limits
\[ 2 < \nu < 3. \]
If a segment of length \(l_{\mathrm{st}}\) is taken as the calculation element, then the number of gaps is
\[ N = \nu N''. \]
If \(q_{\mathrm{el}}(u)\) is the probability of breakdown of an element at voltage \(u\), then the reliability of the winding at this voltage is equal to
\[ R(u) = [1 - q_{\mathrm{el}}(u)]^{N_1}. \tag{2} \]
Let us now determine \(q_{\mathrm{el}}(u)\) and \(N_1\). Since the random quantities \(u_{\mathrm{e}}\) and \(u_{\mathrm{l}}\) are distributed according to different laws, we determine separately \(q(u \ge u_{\mathrm{e}})\) and \(q(u \ge u_{\mathrm{l}})\). The primary statistical material on the breakdown of turn insulation of wires, as well as the scheme of the technological process of applying—
of the enamel coating make it possible to assume that the distribution of \(u_{\mathrm{e}}\) is approximately normal, with density
\[ \varphi_1(u)=\frac{1}{\sqrt{2\pi}\,\sigma_{\mathrm{e}}} \exp\left[-\frac{(u-\overline{u}_{\mathrm{e}})^2}{2\sigma_{\mathrm{e}}^2}\right]. \tag{3} \]
In the limiting case, i.e., in the absence of gaps between insulated conductors, the density of the minimum breakdown voltages of the winding, in accordance with [1, 2], is
\[ \varphi(u)=N_1 f_1(u)\left\{1-\int_{-\infty}^{u} f_1(u)\,du\right\}^{N_1-1} = \]
\[ = N_1 f_1(u)\left[1-F\left(\frac{u-\overline{u}_{\mathrm{e}}}{\sigma_{\mathrm{e}}}\right)\right]^{N_1-1}. \tag{4} \]
\(F(z)\) is the integral distribution function.
The fraction of defective sections of length \(\Delta l\) is
\[ q_{1\mathrm{e}}=\int_{-\infty}^{0} f_1(u)\,du. \tag{5} \]
The probability of a short circuit of a pair is
\[ q_{2\mathrm{e}}=q_{1\mathrm{e}}^2. \tag{6} \]
The probability of an interturn short circuit in the machine is
\[ q=1-(1-q_{1\mathrm{e}}^2)^{N_1}. \tag{7} \]
The probability of the absence of an interturn short circuit, i.e., the lower estimate of the factory reliability of the machine’s turn insulation, is
\[ R_{\mathrm{n}}=(1-q_{1\mathrm{e}}^2)^{N_1}. \tag{8} \]
If a section of standard length is taken as the element, then
\[ R_{\mathrm{n}}=(1-kq_{1\mathrm{st}}^2)^N, \tag{9} \]
where \(k\) is the probability of coincidence of bare spots.
If a twisted pair of length \(l_{\mathrm{st}}\) is taken as the element, then
\[ R_{\mathrm{n}}=(1-q_{\mathrm{twp}})^N, \]
where \(q_{\mathrm{twp}}\) is the fraction of twisted pairs having short circuits.
\[ k=\frac{q_{\mathrm{twp}}}{q_{1\mathrm{st}}^2}. \tag{10} \]
Thus, for calculating the potential accident rate of random-wound windings, it is sufficient to know \(q_{1\mathrm{st}}\), \(N\), and \(k\).
The value \(q_{1\mathrm{st}}\) can be determined from the results of breakdown tests of standard specimens. The value \(k\) can be determined by comparative tests in a shot and in twisted pairs according to formula (10).
Expressions (7) and (9) determine the fraction of machines having interturn short circuits under close contact between insulated conductors, and are an upper estimate of the accident rate of machines for this reason during the running-in period. The presence of varnish and air gaps reduces the accident rate. The distribution of the breakdown voltages of these gaps must obey the exponential law [3], with density
\[ f_2(u)=\lambda \exp(-\lambda u), \tag{11} \]
where
\[ \lambda=\frac{1}{E_{\mathrm{l}}\overline{x}}, \]
\(\overline{x}\) is the mean size of the gap,
\(E_{\mathrm{l}}\) is the electric strength of the impregnating varnish.
When the test voltage is applied, not \(100q\%\) of the machines will break down, but a significantly smaller fraction, determined by the expression
\[ q_{\text{test}}=q\int_{0}^{u_{\text{test}}}\lambda \exp(-\lambda u)\,du =q[1-\exp(-\lambda u_{\text{test}})], \tag{12} \]
where \(u_{\text{test}}\) is the test voltage of the slot.
Thus, although most of the gaps have small dimensions and cannot ensure reliable operation of the machines, they conceal defects in the enamel coating, and therefore the test station reveals only a small fraction of the weak spots.
With time, \(u_l\) decreases for a number of reasons: a decrease in \(E_l\) as a result of heating and mechanical destruction, and a decrease in \(x\) as a result of insufficient cementation. The rate of decrease of \(U_l\) depends on the properties of the impregnating varnish and on the quality of impregnation. It can be shown that, under sufficiently broad assumptions regarding the character of the decrease of \(u_l\), the distribution of time to failure will obey a law close to an exponential one.
\(100q\%\) of the machines have hidden defects with a density of breakdown voltages determined by expression (11). The distribution density of breakdown voltages of the remaining \(100(1-q\%)\) machines is determined by the composition
\[ f_3(u)=\varphi_1(u)*f_2(u)= \]
\[ =\lambda \exp\{-\lambda(u-\overline{u}_{e}-0.5\lambda\sigma_e^2)\} F\left(\frac{u-\overline{u}_{e}-\lambda\sigma_e^2}{\sigma_e}\right). \tag{13} \]
Thus, the distribution density of breakdown voltages in the machine is
\[ q(u)=qf_2(u)+(1-q)f_3(u). \tag{14} \]
During the running-in period, the fraction of machines that failed because of an interturn short circuit is determined by the formula
\[ q_0=\int_{0}^{u_0}q(u)\,du \tag{15} \]
It should be borne in mind that \(u_0\) is sometimes considerably greater than the operating voltage of the slot because of overvoltages that arise when the motor is switched off [4].
\(u_0\) is the solution of the equation
\[ q'(u)=0. \tag{16} \]
The obtained relations were used to estimate the reliability of the turn insulation of an asynchronous motor of type A032-4A. The initial data for the calculation and an assessment of the obtained results are given below. Winding data: wire PЭVA-2 \(\varnothing\) 0.83 mm, \(N=1600\), \(k=0.003\). The statistical parameters of the breakdown voltages of the wire at individual stages of the technological chain are given in Table 1.
Table 1
| Stage | \(0.5\,\overline{U}_e\) | \(0.5\sigma_e\) | \(v=\dfrac{\sigma_e}{\overline{U}_e}\) | \(q_{1\text{st}}\) |
|---|---|---|---|---|
| Before winding on the template | 4.15 | 1.99 | 0.48 | 0.019 |
| After the template | 4.05 | 1.94 | 0.48 | 0.019 |
| After insertion into the slot | 1.40 | 1.18 | 0.84 | 0.120 |
Substituting the given data into formulas (9), (12), and (15), we obtain the following results:
the potential probability of contact \(q = 0.17\);
the probability of failure during the running-in period \(q_0 = 0.065\) \((R = 0.935)\);
the probability of breakdown at the test station \(q_{\mathrm{test}} = 0.0012\).
For comparison we give experimental-statistical data:
the proportion of failures due to turn-to-turn short circuits during the running-in period, according to data from VNIIEM, \(q_0 = 0.02 \ldots 0.055\);
the proportion of machines that had turn-to-turn short circuits at the test station of the Sibelektromotor plant during the period of material selection, \(q_{\mathrm{test}} = 0.0011\).
Conclusions
-
The number and dimensions of exposed areas in the enamel coating determine the number of failures during the running-in period. The failure rate in this period is distributed according to an exponential law with a parameter that depends on the quality of impregnation.
-
The proposed formulas make it possible to estimate the reliability of turn insulation in random-wound windings during the running-in period on the basis of data on the quality of the turn insulation, taking into account damage incurred during winding-insulating operations.
-
The reliability of the winding is only slightly affected by the mean value of the breakdown voltage of the wire turn insulation. The quality of the wires is determined almost exclusively by the scatter in the breakdown-voltage value or by the proportions of defective wires.
-
It has been established that, in the process of laying into slots, a large number of damages are inflicted on the enamel insulation. Reducing wire damage in this operation makes it possible to substantially increase the reliability of random-wound windings when using existing enamelled wires.
-
The level of control of turn insulation at the manufacturing plant is insufficient for detecting weak spots.
References
-
A. S. Zingerman. Determination of breakdown voltage from specimen tests. Elektrichestvo, 1950, No. 3.
-
A. S. Zingerman. Statistical method for determining the breakdown voltage of a dielectric. ZhTF, vol. 18, No. 8, 1948.
-
V. Feller. Introduction to Probability Theory and Its Applications. Mir, 1964.
-
R. Ryudenberg. Transient Processes in Electric Power Systems. IL, 1955.