WEAR OF INSULATION OF INDUCTION MOTOR WINDINGS UNDER RANDOM LOAD

Yu. M. Bashagurov, A. S. Gitman, E. K. Strelbitskii

(Presented by the scientific seminar of the Department of Electrical Machines and by the Electrical Machines Division of the Research Institute of AEM)

An important point in selecting an induction motor for a particular drive is checking it for overheating of the winding. If the load of the drive is a random function of time, then calculation of winding overheating by the method of average losses may lead to erroneous results. Knowledge of the average losses gives no information about the distribution of temperatures, which in certain intervals of time may exceed the permissible temperature for the given insulation class. Because of the exponential dependence of insulation wear on temperature, the resulting error increases as the amplitude of temperature fluctuations increases. In this case, checking by the method of average losses must be supplemented by a check of insulation wear over the calculated service life of the motor.

To determine the additional wear, the problem of excursions of a random function above a given level is used; the initial data are the distributions of the excursion amplitudes \(k\), the durations of the excursions at the given level, and the distribution of pauses \(t_0\) between excursions.

The insulation wear during time \(t\) can be found from the formula

\[ V(t)=\frac{1}{C}\int_0^t e^{b\Theta}\,dt, \tag{1} \]

where \(\Theta\) is the winding temperature; \(b\) and \(C\) are constants for the given insulation class.

For short-time motor overloads, heat transfer may be neglected. Knowing the overheating of the motor winding in the rated mode, \(\Delta\Theta_{\mathrm{nom}}\), at the rated current \(i_{\mathrm{nom}}\), the overheating for current \(i\) can be obtained with satisfactory accuracy as

\[ \Delta\Theta=\left(\frac{i}{i_{\mathrm{nom}}}\right)^2\Delta\Theta_{\mathrm{nom}}. \tag{2} \]

In the general case, when the excursion time is comparable with the time constant of winding heating, allowance for heat transfer is necessary. Let us denote by

\(\kappa=\dfrac{i}{i_{\text{nom}}}\), by \(\kappa_{\text{init}}\) the ratio of the initial current \(i_{\text{init}}\) to the nominal current, and by \(\Theta_{\text{init}}\) the initial temperature. Then the winding temperature during heating, taking into account the ambient temperature \(\Theta_{\text{amb}}\), is

\[ \Theta_{\text{heat}}=\Theta_{\text{amb}}+\Delta\Theta_{\text{nom}}\kappa^2-\Delta\Theta_{\text{nom}}(\kappa^2-\kappa_{\text{init}}^2)e^{-\frac{t}{T}}, \tag{3} \]

and the winding temperature during cooling over time \(t\) is

\[ \Theta_{\text{cool}}=\Theta_{\text{amb}}+\left[\Delta\Theta_{\text{nom}}\kappa^2-\Delta\Theta_{\text{nom}}(\kappa^2-\kappa_{\text{init}}^2)e^{-\frac{t_{\text{h}}}{T}}\right]e^{-\frac{t_{\text{c}}}{T}}, \tag{4} \]

where \(T\) is the time constant of heating and cooling. The additional wear due to the effect of an excursion consists of the wear during heating and cooling

\[ V(t)=V_{\text{heat}}+V_{\text{cool}}. \tag{5} \]

Let us find the terms of formula (5). The insulation wear during heating is

\[ \begin{aligned} V_{\text{heat}} &=\frac{1}{C}\int_{0}^{t_{\text{h}}} e^{b\Theta_{\text{heat}}}\,dt =\frac{1}{C}\int_{0}^{t_{\text{h}}}\exp b\left[\Theta_{\text{amb}}+\Delta\Theta_{\text{nom}}\kappa^2+\right.\\ &\qquad\left.+\Delta\Theta_{\text{nom}}(\kappa_{\text{init}}^2-\kappa^2)e^{-\frac{t}{T}}\right]dt =\frac{1}{C}\exp b(\Theta_{\text{amb}}+\Delta\Theta_{\text{nom}}\kappa^2)\times\\ &\qquad\times\int_{0}^{t_{\text{h}}}\exp\left[b\Delta\Theta_{\text{nom}}(\kappa_{\text{init}}^2-\kappa^2)e^{-\frac{t}{T}}\right]dt . \end{aligned} \tag{6} \]

An integral of the form

\[ \int_{0}^{x} e^{-ae^{-nt}}\,dt\quad (a>0) \]

by the substitution \(-ae^{-nt}=u\) is transformed into an expression in which tabulated integrals of the integral exponential function \(E_i(-u)\) are used:

\[ \int_{0}^{t} e^{-ae^{-nt}}\,dt =\frac{1}{n}\left[E_i(-a)-E_i(-ae^{-nt})\right], \]

therefore

\[ V_{\text{heat}}=\frac{T}{C}\exp b(\Theta_{\text{amb}}+\Delta\Theta_{\text{nom}}\kappa^2)\times \left\{E_i\left[b\Delta\Theta_{\text{nom}}(\kappa_{\text{init}}^2-\kappa^2)\right]- E_i\left[b\Delta\Theta_{\text{nom}}(\kappa_{\text{init}}^2-\kappa^2)e^{-\frac{t_{\text{h}}}{T}}\right]\right\}. \tag{7} \]

The insulation wear during cooling is

\[ \begin{aligned} V_{\text{cool}} &=\frac{1}{C}\int_{0}^{t_{\text{c}}}e^{b\Theta_{\text{cool}}}\,dt =\frac{T}{C}e^{b\Theta_{\text{amb}}} \left\{E_i\left[b\Delta\Theta_{\text{nom}}\kappa^2+\right.\right.\\ &\qquad\left.+b\Delta\Theta_{\text{nom}}(\kappa_{\text{init}}^2-\kappa^2)e^{-\frac{t_{\text{h}}}{T}}\right] -E_i\left[b\Delta\Theta_{\text{nom}}\kappa^2e^{-\frac{t_{\text{c}}}{T}}+\right.\\ &\qquad\left.\left.+b\Delta\Theta_{\text{nom}}(\kappa_{\text{init}}^2-\kappa^2)e^{-\frac{(t_{\text{h}}+t_{\text{c}})}{T}}\right]\right\}. \end{aligned} \tag{8} \]

Using the formulas presented above, a calculation was carried out to determine the additional wear of motors installed on general-purpose lathes, for the following distributions of surge parameters:

\[ f(\kappa)=\frac{1}{\beta-\alpha}=1.37 \qquad \text{for } 1.22<\kappa<1.95 \tag{9} \]

\[ f(\kappa)=0 \qquad \text{for } \kappa<1.22 \text{ and } \kappa>1.95 \tag{10} \]

\[ f(t_b)=0.05\exp(-0.05\,t_b) \tag{11} \]

\[ f(t_0)=0.01\exp(-0.01\,t_0). \tag{12} \]

In this case, the additional wear amounted to about 3% of the wear due to the mean temperature. As the degree of motor loading increases, the share of additional wear increases.