Control of the Specified Reliability of Turn-to-Turn Insulation in Asynchronous Motors During Their Manufacture

CONTROL OF THE SPECIFIED RELIABILITY OF TURN INSULATION IN ASYNCHRONOUS MOTORS DURING THEIR MANUFACTURE

O. P. MURAVLEV, A. D. NEMTSEV, Yu. P. POKHOLKOV

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

The reliability of electrical machines is determined mainly by the reliability of the windings, which depends to a considerable extent on the proper organization of inspection during their manufacture. Inspection of winding insulation during manufacture is carried out at various stages. If the frame and interphase insulation is tested for breakdown by high voltage and weak spots are detected satisfactorily, the situation is quite different with turn insulation. In operation, overvoltages on a phase exceed the rated voltage by a factor of ten or more, whereas during manufacture the turn insulation is tested with a very low voltage. The test voltage for turn insulation in a finished machine is \(1.3 U_{\text{н}}\) per phase, which is not sufficient for detecting weak spots.

Studies carried out by the Tomsk Polytechnic Institute jointly with SKBEM show that, in order to detect hidden defects in turn insulation, it is necessary to apply to a phase a test voltage of magnitude \(0.4 n_{\text{k}}\) (\(kV_{\max}\)), where \(n_{\text{k}}\) is the number of coils in the phase [1]. Unimpregnated stator windings are subjected to the tests.

The purpose of the present work is to provide a method for calculating the control limits of the permissible level of defective turn insulation when tested by high voltage, for a specified reliability of turn insulation \(R_{\text{v zad}}\) during the running-in period.

It is advisable to monitor the reliability \(R_{\text{v zad}}\) by means of control charts based on the mean value of the fraction defective \(\overline{Q}_{\text{br}}\). In constructing the control chart, the main task is to calculate the control limits of the permissible defect level \(Q_{\text{br.v}}\) and \(Q_{\text{br.n}}\). They depend on the characteristics of the production process, and also on the sample size \(n\). The control limits for a control chart based on the mean value of the fraction defective \(\overline{Q}_{\text{br}}\) are calculated from the expression [2]:

\[ Q_{\text{br.v.n.}} = Q_{\text{br}} \pm 3 \sqrt{\frac{\overline{Q}_{\text{br}}(1-\overline{Q}_{\text{br}})}{n}} . \tag{1} \]

The minus sign refers to the lower control limit \(\overline{Q}_{\text{br.n}}\), and the plus sign to the upper limit \(Q_{\text{br.v}}\). If \(Q_{\text{br.n}}\) is negative, then \(Q_{\text{br.n}}=0\) is adopted. After plotting the limits on a diagram, a control chart is obtained (Fig. 1). Along the abscissa axis of a rectangular coordinate system are plotted the numbers of samples of \(n\) stators taken at different moments in time \(1, 2, 3,\ldots\), and along the ordinate axis, graduated from 0 to \(Q_{\text{br.v}}\), are plotted the calculated fractions of defective windings. The exit of a point beyond the pre—

Graph labels: ordinate \(\overline{Q}_{br}\); upper line \(Q_{br.v}=0.147\); lower line \(\overline{Q}_{br}=0.065\).

Fig. 1. Control chart for \(R_{v\,zad}=0.85,\ n=100\)

exceeding the upper control limit \(Q_{br.v}\) indicates that the fraction of rejects is increasing and, consequently, \(R_{v\,zad}\) is decreasing. Conversely, violation of the lower control limit \(Q_{br.n}\) indicates an improvement in quality and entails an increase in reliability. The size of the lot being inspected is determined by the number of manufactured stators, for example, per shift, ten-day period, or month.

On the control chart, the straight line with ordinate \(Q=\overline{Q}_{br}\) is determined by the specified reliability of the turn insulation \(R_{v\,zad}\). Let us consider the determination of \(\overline{Q}_{br}\) from the specified value of \(R_{v\,zad}\). According to [4], the reliability estimate for the turn insulation of random-wound windings as a function of the number of motor starts \(v\) is made from the expression

\[ R_v=[1-q(U_{cv})]^N, \tag{2} \]

where \(N\) is the number of elementary sections of conductor pairs in the motor winding,

\(qU_{cv}\) is the defectiveness of the wire at the maximum possible overvoltage \(U_{cv}\) in operation.

\[ U_{cv}=\frac{U_{fin.v}}{n_c},\ \text{kV}. \tag{3} \]

\(n_c\) is the number of series-connected sections in a phase,

\(U_{fin.v}\) is the amplitude value of the phase voltage, taking into account switching overvoltages over \(v\) motor starts.

\[ U_{fin.v}=\frac{\ln v+0.7}{2.25},\ \text{kV}, \tag{4} \]

\(v\) is the number of motor starts over the calculated period of time.

The value \(qU_{cv}\) is determined from the expression

\[ q(U_{cv})=p^2\lambda\int_{0}^{U_{cv}} F\!\left(\frac{U-\overline{U}_2}{\sigma_2}\right) e^{-\lambda U}\left(1-\frac{U}{U_{cv}}\right)\,dU, \tag{5} \]

where:

\(p\) is the probability of damage to an elementary section of the wire insulation;

\(\lambda\) is the parameter of the exponential distribution law for the breakdown voltage between two damaged conductors;

\(U\) — current value of the voltage, kV;

\(U_2\) — mean value of the breakdown voltage of the defective insulation of a pair of conductors, kV;

\(\sigma_2\) — root-mean-square deviation of the breakdown voltage of the defective insulation of a pair of conductors, kV.

The value of the function

\[ F\left(\frac{U-U_2}{\sigma_2}\right) \]

is determined from the table in Appendix 4 [4].

Thus, knowing the reliability \(R_{\mathrm{v\,zad}}\) calculated for the overvoltage \(U_{\mathrm{c}}\), one can determine the defectiveness \(q(U_{\mathrm{c}})\) from (2), which, for small values of \(q(U_{\mathrm{c}})\), has the form

\[ R_{\mathrm{v}} = 1 - N \cdot q(U_{\mathrm{c}}), \tag{6} \]

then

\[ q(U_{\mathrm{c}})=\frac{1-R_{\mathrm{v\,zad}}}{N}. \tag{7} \]

Knowing the value of the integrand in expression (5) and \(\lambda\), one can determine \(p\), the value of which will be constant for any test voltage, a specified value of \(R_{\mathrm{v\,zad}}\), and a specific machine. Now, knowing \(p\), \(\lambda\), and the value of the integrand found for the upper limit of integration \(U_{\mathrm{c}}\), whose value is determined from (3) for the known test voltage \(0.4 n_{\mathrm{k}}\) kV per phase, one can determine \(q(0.4 n_{\mathrm{k}})\). From the value found for \(q(0.4 n_{\mathrm{k}})\), we calculate \(R'_{\mathrm{v}}\) from (6). The mean reject rate in testing turn insulation with high voltage is

\[ \overline{Q}_{\mathrm{br}} = 1 - R'_{\mathrm{v}} . \tag{8} \]

From this value \(\overline{Q}_{\mathrm{br}}\), the control limits for the control chart are calculated according to (1). Since, when turn insulation is tested with high voltage, stators whose insulation would have broken down under an overvoltage of \(0.4 n_{\mathrm{k}}\) kV max occurring in operation are rejected, the reliability of the stators that have passed the control tests at this voltage will be equal to

\[ R''_{\mathrm{v}} = -\frac{R_{\mathrm{v\,zad}}}{1-Q_{\mathrm{br}}}, \tag{9} \]

The reliability of the stators released, \(R''_{\mathrm{v}}\), proved to be higher than the specified \(R_{\mathrm{v\,zad}}\). In order to obtain the specified reliability after high-voltage testing, the calculation must use not the specified reliability value \(R_{\mathrm{v\,zad}}\), but the calculated value

\[ R_{\mathrm{v}}^{*}=K' \cdot R_{\mathrm{v\,zad}} . \tag{10} \]

Approximate values of the coefficients \(K'\) as a function of \(R_{\mathrm{v\,zad}}\) for a test voltage of \(2.5/2\) kV are given in Table 1.

Table 1

Thus, having determined the value \(\overline{Q}_{\mathrm{br}}\) from the specified reliability, we construct a control chart with limits \(Q_{\mathrm{br\text{-}v}}\) and \(Q_{\mathrm{br\text{-}n}}\). If, during control, the points lie within the control limits, then, when asynchronous motors are released, the specified reliability of the turn insulation \(R_{\mathrm{v\,zad}}\) is guaranteed. If the points go beyond the upper limit \(Q_{\mathrm{br\text{-}v}}\), then it is necessary to look for the causes producing a deterioration of the manufacturing process of manufactur-

of the windings. An example of constructing a control chart for an AO2-32-4 motor.

Initial data:

\[ R_{\mathrm{v\,set}}=0.85;\quad t_{\mathrm{calc}}=5000\ \text{h};\quad f_{\mathrm{incl}}=30\ \text{1/h}; \]
\[ l_{\mathrm{w}}=60\ \text{mm};\quad \bar r_{\mathrm{test}}=1.527\ \text{mm};\quad \bar E'_{\mathrm{v}}=1.99\ \text{kV/mm}; \]
\[ E_n=42.42\ \text{kV/mm}\ [4]. \]

The intermediate values required for calculating \(q(0.4 n_k)\) were calculated from the corresponding formulas in [4] and are given below:

\[ \overline U_1=0.656\ \text{kV};\quad \overline U_2=1.18\ \text{kV};\quad \sigma_1=0.372\ \text{kV}; \]
\[ \sigma_2=0.525\ \text{kV};\quad N=1.53\cdot 10^6;\quad \lambda=0.295\ \text{1/kV}; \]
\[ \nu=150000;\quad U_{\phi\cdot \mathrm{ph}\cdot \nu}=5.64\ \text{kV};\quad U_{c\nu}=0.94\ \text{kV}; \]
\[ U'_{c\nu}=0.589\ \text{kV}; \]

The probability of breakdown of an “elementary section” of the insulation of a pair of conductors that had damage before impregnation, over a period of 5000 h, is

\[ q(U_{c\nu})= \frac{1-R_{\mathrm{v}}^*}{N} = \frac{1-0.85\cdot 0.945}{1.53\cdot 10^6} = 0.129\cdot 10^{-6}. \]

The probability of damage to an “elementary section” of the insulation of a conductor is

\[ p^2= \frac{q(U_{c\nu})} { U_{c\nu} \lambda \int_{0}^{\infty} F\!\left(\frac{U-\overline U_2}{\sigma_2}\right) e^{-\lambda U} \left(1-\frac{U}{U_{c\nu}}\right) } = \frac{0.129\cdot 10^{-6}}{0.295\cdot 0.0265} = 16.5\cdot 10^{-6}. \]

From (5) we determine \(q(U'_{c\nu})\), substituting \(U'_{c\nu}=0.589\ \text{kV}\) as the upper limit of integration,

\[ q(U'_{c\nu})=16.5\cdot 10^{-6}\cdot 0.295\cdot 0.00945=0.046\cdot 10^{-6}. \]

The value of the integrand function \(f(U)dU=0.00945\). From the found value \(q(U'_{c\nu})\), by (6) we calculate \(R'_{\mathrm{v}}\)

\[ R'_{\mathrm{v}}=1-1.53\cdot 10^6\cdot 0.046\cdot 10^{-6}=0.9296. \]

The mean value of rejects when turn insulation is tested with high voltage is determined from (8):

\[ Q_{\mathrm{rej}}=1-0.9296=0.0704. \]

The calculation of the control limits for the control chart is performed according to (1). The value of the upper control limit \(Q_{\mathrm{rej.v}}\) for different specified values of \(R_{\mathrm{v\,set}}\) and sample sizes \(n\) is presented in Table 2. Table 3 gives the values of \(Q_{\mathrm{rej}}\) for different values of \(R_{\mathrm{v\,set}}\). The reliability of stators that have passed the high-voltage control tests is determined from (9), taking (10) into account.

\[ R''_{\mathrm{v}}= \frac{R'_{\mathrm{v}}}{1-\overline U_{\mathrm{b}}} = \frac{0.85\cdot 0.945}{1-0.0704} = 0.865. \]

Conclusion

A method has been proposed for controlling the specified reliability of turn insulation \(R_{\mathrm{v\,set}}\) during the run-in period by means of control charts, the parameters of which are determined directly from the specified reliability value.

Table 2

Table 3

REFERENCES

1. Yu. P. Pokholkov. The influence of winding-insulation work and insulation quality on the reliability of windings of electrical machines. Candidate’s dissertation, TPI, Tomsk, 1966.

2. R. Shtorm. Probability Theory. Mathematical Statistics. Statistical Quality Control. Mir, 1970.

3. D. Cowden. Statistical Methods of Quality Control. Fizmatgiz, 1961.

4. Three-phase asynchronous motors for general use, rated from 0.12 to 100 kW. Calculation of the reliability of random windings. R.T.M. Tomsk, 1970.