Accounting for Electrodynamic Forces Between Conductors in Assessing the Reliability of Turn Insulation in Induction Motor Windings During the Run-In Period

V. I. Derevyanko, Yu. P. Pokholkov

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

The reliability and service life of induction motors depend to a great extent on the operating conditions of the winding insulation and on technological, design, and operational factors [1]. The main factors acting on the winding insulation of electrical machines in operation are known: heat, commutation overvoltages, humidity, vibrations, and mechanical forces arising during motor starts and reversals. A quantitative assessment of all these factors would make it possible not only to predict motor reliability with greater accuracy, but also to select optimal winding design variants when designing motors for particular operating conditions.

In the present work an attempt is made to quantitatively assess the influence of electrodynamic forces arising between round conductors during motor switching-on and reversals on the reliability of winding turn insulation.

Solving this problem is especially important for motors of larger frame sizes, in which the starting current and short-circuit current reach high values. The magnitude of the electrodynamic forces between conductors of circular cross section, according to [2], may be estimated by the formula

\[ F_{1/2}=1.02\left(\frac{i_p}{10}\right)^2\cdot \frac{l}{a}\ \text{kgf}, \tag{1} \]

where

\[ i_p \]
is the starting current, kA;

\[ l \]
is the length of the interacting conductors, mm;

\[ a \]
is the distance between the centers of the conductors, mm.

Notes. We compare the obtained value \(F_{1/2}\), under the action of which displacement of conductors relative to one another occurs, cemented by an impregnating compound, with the force \(F_{\mathrm{c}}\), where \(F_{\mathrm{c}}\) is the adhesion force of impregnated conductors, depending on the type of varnish being used for impregnation. If \(F_{1/2}<F_{\mathrm{c}}\), it is not advisable to take into account the influence of electrodynamic forces on the reliability of winding turn insulation.

The influence of electrodynamic forces on the reliability of winding turn insulation was assessed on the basis of the following assumptions:

1. The distribution of angles between the axes of conductors in a random winding obeys an exponential law.

2. Mechanical destruction of conductor insulation under the action-

the action of electrodynamic forces occurs only at points of close contact of the conductors in the outer layers of the end-winding section and has the character of abrasion.

1. Electrodynamic forces between conductors are significant only when starting currents and short-circuit currents flow; therefore, the number of cycles of abrasion of the wire insulation is equal to the number of motor switch-ons.

2. The cementing ability of the impregnating compound does not change during the run-in period.

The assumptions listed make it possible to quantitatively estimate the probability of failure of the turn insulation of windings caused by the action of electrodynamic forces between conductors.

1. The parameter of the exponential distribution between the axes of the conductors, \(\lambda\):

\[ \lambda = \frac{1}{\bar{\gamma}^{0}},\; 1/\mathrm{deg}, \tag{2} \]

where

\[ \bar{\gamma}^{0} \]

is the mean angle between the axes of the conductors.

1. The number of conductors in the outer layer of the winding section

\[ S_{\mathrm{out}} = \frac{\Pi_c}{d_{\mathrm{ins}\,\max}}, \tag{3} \]

where

\[ d_{\mathrm{ins}\,\max} = 1.075 \cdot d_{\mathrm{ins}} \cdot \sqrt{\frac{1}{k_{\mathrm{fill}}}}; \]

\(d_{\mathrm{ins}}\) — diameter of the insulated wire, mm,
\(k_{\mathrm{fill}}\) — slot fill factor;
\(\Pi_c\) — perimeter of the section, mm.

1. The number of conductors in the inner layers of the section

\[ S_{\mathrm{in}} = S - S_{\mathrm{out}}. \tag{4} \]

1. The number of elementary segments of conductor pairs subjected to the action of electrodynamic forces [EDF]

\[ N = \left(1 - e^{-\frac{\sqrt{k_{\mathrm{fill}}}}{1.075}}\right) \left(S_{\mathrm{out}} + 1.5S_{\mathrm{in}} - 1.5\right) \times \frac{k \cdot z \cdot l_w}{l_{\mathrm{el}}} \cdot k_y, \tag{5} \]

where

\(k\) — number of layers in the slot;
\(z\) — number of stator slots;
\(l_w\) — mean length of a winding turn, mm,
\(k_y\) — coefficient taking into account the part of the conductors capable of moving relative to one another;

\[ k_y = \frac{S_{\mathrm{out}}}{S} \cdot \frac{l_{\mathrm{end}}}{l_w}; \]

\(l_{\mathrm{el}}\) — length of the elementary segment of the conductor, mm.

1. The number of elementary segments of conductor pairs characterized by a definite mean angle \((\bar{\gamma}_i)\) between the axes of the conductors.

\[ k_i = N \cdot \lambda \cdot e^{-\lambda \bar{\gamma}_i} \tag{6} \]

In the calculation, the following values of \(\bar{\gamma}_i\) may be used: \(2^\circ, 4^\circ, 6^\circ, 8^\circ, 10^\circ, 12^\circ\).

1. The level of electrodynamic force between elementary sections of conductors positioned relative to one another at an angle \(\overline{\gamma_i}\):

\[ F_i=\left[1.02\left(\frac{i_p\cdot k_a}{10}\right)^2\frac{2\cdot l_{\mathrm{el}}}{d_{\mathrm{iz}}}\right]\cos\overline{\gamma_i},\ [kg] \tag{7} \]

1. The number of machine switch-ons during the time \(t_{\mathrm{calc}}\), \([hours]\),

\[ \nu=f_{\mathrm{sw}}\cdot t_{\mathrm{calc}}, \tag{8} \]

where

\(f_{\mathrm{sw}}\) — switching frequency, \([1/hour]\).

1. The probability of destruction of the insulation of an elementary section of a pair of conductors under the action of the force \(F_i\) over \(\nu\) machine switch-ons

\[ P_i(F_i)=F\left(\frac{\nu-\overline{\nu_i}}{\sigma_{\nu_i}}\right), \tag{9} \]

where

\(\overline{\nu_i}\) — the mean number of actions of the force \(F_i\) causing insulation failure within the elementary section;

\(\sigma_{\nu_i}\) — the root-mean-square deviation of the number of actions of the force \(F_i\) causing failure;

\(\overline{\nu_i}\) and \(\sigma_{\nu_i}\) are determined experimentally.

Tests of wires for the purpose of determining the number of cycles to complete abrasion of the insulation as a function of the applied load were carried out on the SE-68 apparatus.

On the basis of the experimental data, the dependence \(\ln\nu=f(F_i)\) was constructed. This function has the equation of a broken line consisting of two straight lines intersecting at the point \(F_{1/2}=150\ g/mm\). The equation of the straight line in the region in which the working values of the electrodynamic force lie may be represented in the form of Table 1.

\[ \ln\nu=12.9-3.88\cdot10^{-2}. \tag{10} \]

Table 1

Number of cycles to failure of glass-fiber insulation

The obtained dependence \(\ln\nu=f(F_i)\) may be used to evaluate the effect of electrodynamic forces on the insulation of random-wound motor windings.

1. The probability of destruction of a group of elementary sections subjected, during motor switch-ons, to the action of the force \(F_i\),

\[ P_{ki}(F_i)=1-\left[1-P_i(F_i)\right]^{ki}. \tag{11} \]

1. The probability of failure of the turn insulation of the winding as a result of the action of electrodynamic forces between the winding conductors

\[ P_{\mathrm{w}}(F_i)=1-\prod_{i=1}^{i}\left[1-P_{ki}(F_i)\right], \tag{12} \]

where

\(i\) — the number of groups of elementary sections characterized by a definite angle between the axes of the conductors.

According to item 5, \(i=7\).

11. Reliability of the turn insulation of the winding, caused by the action of EDF between conductors

\[ R_{\mathrm{v}}(F_i)=1-P_{\mathrm{obl}}(F_i). \tag{13} \]

According to this model, the electrodynamic forces between conductors of various machines of the VAO series with fibrous insulation were calculated (the results are summarized in Table 2), as was the reliability of the turn insulation

Table 2

Calculated EDF data \((\mathrm{g}/\mathrm{mm})\) between conductors of the VAO series

of the VAO-72-4, which proved to be equal to 0.404 at 50,000 thousand starts of the machine. The overall reliability of the turn insulation, taking into account commutation overvoltages and EDF, proved to be equal to

\[ R_b(\gamma,F_i)=0.635\cdot 0.404=0.256, \]

the magnitude of which approximately corresponds to the experimental reliability for this machine and \(\gamma=50\,000\), obtained at the Kuzbass-Elektromotor plant.

LITERATURE

1. RTM for three-phase induction motors with power ratings of 0.18–100 kW. Calculation of the reliability of random-wound windings. Tomsk, TsBTI, 1970.

2. G. B. Kholyavskii. Calculation of electrodynamic forces in electrical apparatus. Gosenergoizdat, 1962.