CALCULATION OF STATOR WINDING OVERHEATING IN THE DESIGN OF AN OPTIMAL INDUCTION MOTOR

D. I. Sannikov, V. A. Zhadan

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

When designing an optimal induction motor with the use of an electronic digital computer, the excess temperature of the stator winding, which is one of the constraints, is usually determined by the widely used formula

\[ \theta_{M_1}=\frac{P_{\mathrm{gr}}}{S_0\alpha_0}, \tag{1} \]

where \(P_{\mathrm{gr}}\) is the heating losses,

\[ S_0=\pi D_j l+0.5\pi D_j^2 \]

is the surface area of the active part,

\(D_j\) and \(l\) are the outside diameter and length of the stator,

\(\alpha_0\) is a conventional heat-transfer coefficient, equal to the density of the heat flux through a unit of surface \(S_0\), referred to one degree of the average overheating of the stator winding. Formula (1) and the coefficient \(\alpha_0\), strictly speaking, do not correspond to the physical picture of heat removal from the motor; however, the relative stability of \(\alpha_0\) when the dimensions and losses of the motor vary within ordinary limits makes it possible to choose its value on the basis of experimental heating data for existing machines and to use formula (1) for calculating the heating of newly designed motors.

However, a change in the parameters of the cooling system in a new motor substantially affects the coefficient \(\alpha_0\). Therefore, for design purposes, a computational method is needed for determining \(\alpha_0\) with allowance for these changes before tests of a prototype of the new motor.

On the basis of (1)

\[ \alpha_0=\frac{P_{\mathrm{gr}}}{\theta_{M_1}S_0}. \tag{3} \]

In enclosed fan-cooled induction motors, the overheating of the winding is composed of temperature drops in the sections through which the heat flux passes successively from the winding to the surrounding air. The thermal resistance of this path may be divided into an external component, dependent on the cooling conditions of the frame, and an internal component, independent of them. On the basis of the usual method of thermal calculation, expression (3) is reduced to the form

\[ \alpha_0=\frac{1}{\rho_1+\rho_2}, \tag{4} \]

where \(\rho_1\) and \(\rho_2\) are, respectively, the external and internal specific thermal resistances

\[ \rho_1=\left(1+\frac{D_j}{2l}\right)\frac{1+\partial_{\mathrm{st}}}{\alpha_{\Sigma}\left(C_0+\frac{C_1}{l}\right)}, \tag{5} \]

\[ \rho_2=\left(1+\frac{D_j}{2l}\right)\left[ k_{\Delta}\rho_{\Delta}+k_F\left(\rho_j+\frac{\rho_z}{3}\right) +k_M\left(\rho_i+\rho_M\frac{l}{l+l_l}\right) \right]. \tag{6} \]

Here \(\alpha_{\Sigma}\) is the heat-transfer coefficient of the frame;
\(C_0\) and \(C_1\) are parameters depending on the frame design;
\(\partial_{\mathrm{st}}\) is the relative increase in resistance due to nonuniform heating of the bedplate, amounting on average to \(7\%\)

\[ k_{\Delta}=\frac{P_{\Delta}}{P_{\mathrm{gr}}},\qquad k_F=\frac{P_F}{P_{\mathrm{gr}}},\qquad k_M=\frac{P_{M1}}{P_{\mathrm{gr}}}. \tag{7} \]

Here \(P_{\Delta}\) and \(P_F\) are the heat flows through the joint between the stator and the bedplate and through the stator core.

\(P_{M1}\) is the loss in the stator winding.
The heating losses \(P_{\mathrm{gr}}\) in this method, for enclosed motors, are taken to be equal to the total losses excluding \(2/3\) of the mechanical losses, with the stray losses taken equal to \(2\%\) of the input power.

\(\rho_{\Delta}, \rho_j, \rho_z, \rho_i, \rho_M\) are the specific thermal resistances of the joint between the stator and the bedplate, of the yoke, of the stator teeth, of the winding insulation in the slot portion, and of the winding conductors in the longitudinal direction; the general expression is

\[ \rho=R\pi D_j l, \tag{8} \]

where \(R\) is the thermal resistance of the corresponding section according to the thermal-calculation method.

When designing the motor, the outside diameter of the stator \(D_j\) is selected initially, while the length \(l\) and the remaining dimensions of the active part are determined during the design process. Therefore, knowing \(D_j\) and having approximately determined the overhang length of the end turns, it is possible, before the start of the electromagnetic calculation, to develop the frame design and determine those dimensions of the bedplate that do not depend on \(l\), as well as the dimensions of the bearing shield:

\[ D_{\mathrm{st}} \]
—the outside diameter of the bedplate at the base of the ribs,

\[ N_p \text{ and } N_p' \]
—the number of normal and shortened ribs with heights, respectively, \(h_p\) and \(h_p'\),

\[ b_0 \text{ and } b_{\mathrm{sr}} \]
—the rib width at the base and at mid-height,

\[ N_p \text{ and } N_l \]
—the number of longitudinal projections on the bedplate and of feet,

\[ h_{\mathrm{sh}} \]
—the shield overhang,

\[ \Delta l=L_{\mathrm{st}}-l_p \]
—is selected as a function of the overhang of the end turns, while the bedplate length \(L_{\mathrm{st}}\) and the stator \(l\) are still unknown,

\[ \partial l_p=L_{\mathrm{st}}-l_p, \]
where \(l_p\) is the mean rib length.

After this, the fan and shroud are designed, and from the ventilation calculation the cooling-air velocity \(V_p\) is determined, which makes it possible to calculate the heat-transfer coefficient of the frame \(\alpha_{\Sigma}\), and then the rib efficiency coefficient \(k_p\). The calculation of the coefficients \(C_0\) and \(C_1\) is carried out by the following formulas:

\[ \partial L=0.5D_{\mathrm{st}}+h_{\mathrm{sh}}, \tag{9} \]

\[ \Pi_p'=1.95\left(N_p h_p+N_p' h_p'\right)k_p, \tag{10} \]

\[ \partial \Pi = 2(N_l + 0.8N_i)h_p - (N_p + N'_p)b_0, \tag{11} \]

\[ k_c = \frac{1}{D_j}\left(D_{st} + \frac{\partial \Pi + \Pi'_p}{\pi}\right), \tag{12} \]

\[ L' = \frac{1}{D_j}\left(D_{st}\,\partial L - \frac{\Pi'_p\,\partial l_p}{\pi}\right), \tag{13} \]

\[ C_0 = 0.615k_c - 0.385\frac{L'}{D_{st}}, \tag{14} \]

\[ C_1 = k_c\Delta l + L' + 0.385(D_{st} - \Delta l)\left(k_c + \frac{L'}{D_{st}}\right). \tag{15} \]

According to (4--6), \(a_0\) is a function of the unknown dimension \(l\); however, calculation examples show that this dependence is very weakly expressed. Thus, when \(l\) changes by 25%, \(a_0\) changes by no more than 2.5%. Therefore, when calculating \(a_0\), the length \(l\) may initially be specified with an accuracy of \(\pm 10\%\).

Likewise, changing within fairly wide limits (\(\pm 25\%\)) such parameters as the height of the stator back and the tooth width, with the corresponding redistribution of specific loads or without it, leads to a change in \(a_0\) within the range 1.5--5%. Redistribution of losses without changing the geometry of the active part has a somewhat greater effect.

Thus, inaccuracies in the preliminary selection of the coefficients \(k\) and \(\rho\) have little effect on the result of the calculation. Recommended values of these coefficients for enclosed low-power induction motors are given in Table 1 as a function of the number of poles. Use of these values leads to the simplified formula

\[ a_0 = \frac{1} {\left(1 + \frac{D_j}{2l}\right) \left[ \frac{1.07}{a_z(C_0 + C_1/l)} + A + BD_j \right]} . \tag{16} \]

The coefficients \(A\) and \(B\) are given in Table 1.

Table 1

Mean values of the coefficients in expressions (6) and (16)
for motors of frames 1--4 of the AO2 and AO4 series

Table 2

Results of the thermal calculation of motors and their comparison with experiment

It is seen from Table 2 that the results of the calculation by formula (16) have a discrepancy with experiment $\Delta \Theta_{\mathrm{m}}$ that is admissible for approximate calculations. Thus, the proposed method can be used in preparing the initial data for designing induction motors on an electronic digital computer.

To improve the accuracy of the calculation of the individual coefficients $k$ and $\rho$, the computation of $\alpha_0$ may be included in the program for the computer.