METHOD FOR THERMAL CALCULATION OF LOW-POWER ENCLOSED ASYNCHRONOUS MOTORS

V. A. Zhadan, Yu. V. Kopylov, D. I. Sannikov

(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)

Determination of the average temperature of the motor winding may be carried out without taking into account the asymmetry of its cooling. In this case, the paths of heat transfer from the internal parts of the motor to the air flowing over the surface of the frame are represented by the thermal circuit in Fig. 1. The main path of the heat flow passes from the rotor to the frame through the thermal resistances of the air gap \(R_{\delta}\), the stator teeth \(R_z\), the yoke \(R_j\), and the joint between the stator and the frame \(R_{\Delta}\); the resistance \(R'_{\text{st}}\) takes into account the nonuniform heating of the frame. This flow is composed of part of the losses in the rotor \(P_r\), the losses in the teeth and yoke of the stator \(P_z\) and \(P_j\), and the heat flow from the stator winding through the insulation resistance in the slot portion \(R_i\).

Fig. 1

In view of the considerable nonuniformity of winding heating along its length, it is necessary to take into account the thermal resistance between the end and slot portions along the conductors \(R_{\ell p}\), and to divide the copper losses \(P_{m1}\) into two heat sources: \(P_{mp}\) and \(P_{m\ell}\).

In addition to the main path, part of the heat is removed from the end portions of the rotor and stator to the air inside the frame and from it to the frame. The summar-

... the heat flow formed by the heating losses of the motor is transferred from the surface of the enclosure to the outside air, thereby creating a temperature difference; in the diagram this is taken into account by the thermal resistance \(R_{\alpha}\).

The calculation of a two-loop thermal circuit involves considerable difficulties; however, for practical calculations it can be substantially simplified.

First, on the basis of experiments it has been established that in enclosed motors ventilated from the inside only by the rotor fan blades and having no grille in the end parts of the stator, the magnitude of the temperature difference from the end parts to the slot portion ensures transfer into the slot portion of the winding, owing to the thermal conductivity of copper, of 85--98% of the losses in the end parts. For example, in motors A02-42-4 this difference is 5.2--5.6°C, which, for the calculated value \(R_{\text{ep}}=0.035\ \mathrm{deg/W}\), corresponds to a heat flow of 150--160 W, whereas the losses in the end parts are on average equal to 166 W. This circumstance makes it possible to neglect heat removal from the stator through the internal air.

Second, one may specify the ratio of the heat-flow density through the lateral and end surfaces of the rotor, in view of the fact that, according to experience, the temperature differences rotor--stator core and rotor--internal air differ little from each other. On average, the heat-flow density on the end surface, where the air is strongly turbulized by the fan blades, is twice as large as on the closed cylindrical surface in the air gap. Consequently, the heat flow from the rotor to the stator (through \(R_{\delta}\)) can be calculated preliminarily as

\[ P_{\delta} = (P_{M2} + P_{\partial})\,\frac{S_{\delta}}{S_{\delta}+2S_T}. \tag{1} \]

Here

\(P_{M2}\) -- losses in the rotor winding,

\(P_{\partial}\) -- additional losses attributed to the rotor, which in the thermal calculation are taken equal to 2% of the input power.

\[ S_T = 2(2b_k + h_k)\pi(D-h_k) + 4n_l h_l b_l \tag{2} \]

is the area of the end surfaces,

\(n_l\) -- number of rotor blades on one side,

\[ S_{\delta}=\pi D l_1 \tag{3} \]

is the area of the air gap.

The necessary dimensions are given in Fig. 2.

Thus, the thermal circuit is converted into a single-line circuit, in all sections of which the heat flows are known. In particular, the flow through the joint \((R_{\Delta})\)

\[ P_{\Delta}=P_{M1}+P_z+P_j+P_{\delta}. \tag{4} \]

The total heating losses are

\[ P_{\text{hr}}=P_{M1}+P_{M2}+P_z+P_j+P_o+\frac{1}{3}P_{\text{mech}}. \tag{5} \]

The mechanical losses \(P_{\text{mech}}\) are taken into account only partially, since the losses in the fan and bearings have little effect on the heating of the motor.

To determine the average overheating of the stator winding, it is sufficient to calculate the thermal resistances, except for those indicated by a dashed line.

\[ R_{\text{lp}}=\frac{l_1+l_l}{3\pi z_1 n_p d^2\lambda_{\mathrm{Cu}}}. \tag{6} \]

Fig. 2

Here \(z_1\) is the number of stator teeth,
\(n_{\mathrm{p}}\) is the number of elementary conductors in a slot,
\(d\) is the diameter of the wire without insulation,
\(\lambda_{\mathrm{Cu}} = 3.85\ \mathrm{W}/\mathrm{cm\,deg}\) is the specific thermal conductivity of copper.

\[ R_{\mathrm{i}}= \frac{1}{z_1 l_1} \left[ \frac{\Delta_{\mathrm{i}}'}{\lambda_{\mathrm{i}}\Pi} + \frac{1}{70\lambda_{\mathrm{p}}} \sqrt{ \frac{d'}{d} \left( \frac{2}{\sqrt{k_{\mathrm{z}}}}-1 \right) -1 } \right]. \tag{7} \]

Here

\[ \Delta_{\mathrm{i}}'=\Delta_{\mathrm{i}}+0.005\ \mathrm{cm} \]

is the calculated thickness of the slot insulation,

\[ \Pi = 2h_{\mathrm{p}}+\frac{d_1}{2}-\frac{d_2}{4} \tag{8} \]

is the perimeter of the oval-shaped slot; for a trapezoidal slot, \(a_1\) is taken instead of \(d_1/2\),

\(d'\) is the diameter of the insulated wire,

\[ k_{\mathrm{z}}=\frac{n_{\mathrm{p}}d'^2}{S_{\mathrm{p}}'} \]

is the slot fill factor,

\(S_{\mathrm{p}}'\) is the area of the slot without insulation and wedge;

specific thermal conductivities:

\[ \lambda_{\mathrm{i}}=0.0010\ \mathrm{W}/\mathrm{cm\,deg} \]

for insulation based on electrical cardboard, and \(0.0016\) for insulation based on asbestos,

\[ \lambda_{\mathrm{p}}=0.0014 \]

for conductor insulation.

\[ R_z=\frac{h_{\mathrm{p}}}{z_i\cdot l_1 b_z\cdot k_{\mathrm{Fe}}\lambda_{\mathrm{Fe}}}, \tag{9} \]

\(k_{\mathrm{Fe}}\) is the steel filling factor,

\(\lambda_{\mathrm{Fe}}=0.41\ \text{W}/\text{cm deg}\) is the thermal conductivity of E11 or E12 steel along the sheet.

\[ R_j=\frac{h_j}{\pi(D_j-h_j)l_1 k_{\mathrm{Fe}}\lambda_{\mathrm{Fe}}}, \tag{10} \]

\[ R_{\Delta}=\frac{\Delta_{\mathrm{v}}}{\pi\cdot D_j\cdot l_1\cdot \lambda_{\mathrm{v}}} =\frac{1.2\cdot 10^{-4}}{\pi l_1 \lambda_{\mathrm{v}}}, \tag{11} \]

\(\Delta_{\mathrm{v}}=1.2\cdot 10^{-4}\cdot D_j\) is the calculated gap between the stator stack and the frame,

\(\lambda_{\mathrm{v}}=2.9\cdot 10^{-4}\) is the thermal conductivity of air.

The total axial thermal resistance of the frame is

\[ R_{\mathrm{st}}=\frac{L_{\mathrm{st}}}{S_{\mathrm{st}}\lambda_{\mathrm{st}}}, \tag{12} \]

\(\lambda_{\mathrm{st}}=0.62\) is the thermal conductivity of cast iron;

\[ S_{\mathrm{st}}=\pi(D_{\mathrm{st}}-h_{\mathrm{st}})h_{\mathrm{st}} +(N_{\mathrm{p}}h_{\mathrm{p}}+N_{\mathrm{p}}'h_{\mathrm{p}}')b_{\mathrm{p}} +(0.5N_{\mathrm{l}}+0.8N_{\mathrm{p}})h_{\mathrm{p}}^2 \tag{13} \]

is the cross-sectional area of the frame.

Here

\(N_{\mathrm{p}}\) is the number of ribs of normal height \(h_{\mathrm{p}}\),

\(N_{\mathrm{p}}'\) and \(h_{\mathrm{p}}'\) are the number and height of shortened ribs,

\(N_{\mathrm{l}}\) and \(N_{\mathrm{p}}\) are the number of feet and longitudinal bosses on the frame,

\[ h_{\mathrm{st}}=\frac{D_{\mathrm{st}}}{2}-\frac{D_j+D_j'}{4} \tag{14} \]

is the mean thickness of the frame.

The thermal resistance from the surface of the shell to the air, taking into account all types of heat transfer, as well as the effect of heating of the air flow, is calculated on the basis of the transformed criterion heat-transfer formula of the form

\[ \mathrm{Nu}=C\mathrm{Re}^m. \]

The effective blowing velocity is

\[ v_{\mathrm{eff}}=\sqrt{v_{\mathrm{p}}^2+\frac{v_{\theta}^2}{25}}, \tag{15} \]

where \(V_{\mathrm{o}}\) and \(V_{\mathrm{p}}\) are the circumferential velocity of the fan and the average air velocity at the exit from under the cover, obtained from the ventilation calculation (in m/s).

The total heat-transfer coefficient is

\[ \alpha_{\mathrm{e}}=9.5\cdot 10^{-4}\frac{v_{\mathrm{eff}}^{0.65}}{D_{\mathrm{st}}^{0.35}} \tag{16} \]

(here \(D_{\mathrm{st}}\) is expressed in meters).

The surface of the shell without ribs is

\[ S_{\mathrm{gl}}=[\pi D_{\mathrm{st}}+2(N_{\mathrm{l}}+0.8N_{\mathrm{p}})h_{\mathrm{p}}]L_{\mathrm{st}}- \]

\[ -(N_{\mathrm{p}}+N_{\mathrm{p}}')b_0 l_{\mathrm{po}}+\frac{\pi}{2}D_{\mathrm{st}}^2+3D_{\mathrm{st}}h_{\mathrm{sh}}. \tag{17} \]

The perimeter of a rib is

\[ P_{\mathrm{p}}=2h_{\mathrm{p}}+1.14R_{\mathrm{v}}-0.86R_0. \tag{18} \]

Rib surface area

\[ S_{\mathrm{p}} = N_{\mathrm{p}} l_{\mathrm{p}} \Pi_{\mathrm{p}} + N'_{\mathrm{p}} l'_{\mathrm{p}} \Pi'_{\mathrm{p}} . \tag{19} \]

Rib efficiency coefficient

\[ k_{\mathrm{p}} = \frac{1}{1 + 0.6 \dfrac{h_{\mathrm{p}}^{2}\alpha_{\Sigma}}{\lambda_{\mathrm{ct}} b_{1/3}}}. \tag{20} \]

Frame length coefficient

\[ k_{1} = 1 + 0.385 \left( \frac{L_{\mathrm{ct}}}{D_{\mathrm{ct}}} - 1 \right). \tag{21} \]

External thermal resistance

\[ R_{\alpha} = \frac{k_{1}}{\alpha_{\Sigma}(S_{\mathrm{r.l}} + k_{\mathrm{p}} S_{\mathrm{p}})} . \tag{22} \]

Temperature drops across the individual resistances

\[ \theta_{\mathrm{lp}} = P_{\mathrm{m.l}} \cdot R_{\mathrm{lp}} = P_{\mathrm{m1}} \frac{l_{\mathrm{l}}}{l_{1} + l_{\mathrm{l}}} R_{\mathrm{lp}}, \tag{23} \]

\[ \theta_{\mathrm{i}} = P_{\mathrm{m1}} R_{\mathrm{i}}, \tag{24} \]

\[ \theta_{z} = \left( \frac{1}{3} P_{\mathrm{m1}} + \frac{1}{3} P_{z} + \frac{1}{2} P_{\delta} \right) R_{z}, \tag{25} \]

\[ \theta_{j} = \left( P_{\mathrm{m1}} + P_{z} + P_{\delta} + \frac{1}{2} P_{j} \right) R_{j}, \tag{26} \]

\[ \theta_{\Delta} = P_{\Delta} R_{\Delta}, \tag{27} \]

\[ \theta'_{\mathrm{ct}} = \frac{(1 - \varepsilon)(P_{\Delta} - \varepsilon P_{\mathrm{gr}})} {\dfrac{1}{4R_{\alpha}} + \dfrac{12}{R_{\mathrm{ct}}}} \tag{28} \]

is the excess temperature of the middle part of the frame, with relative length

\[ \varepsilon = \frac{l_{1}}{L_{\mathrm{ct}}} \tag{29} \]

above the average temperature of the entire shell

\[ \theta_{\alpha} = P_{\mathrm{gr}} R_{\alpha}. \tag{30} \]

Since the mean steady-state temperature of the stator winding is equal to

\[ \theta_{\mathrm{m1}} = \frac{\theta_{\mathrm{mp}} l_{1} + \theta_{\mathrm{m.l}} l_{\mathrm{l}}} {l_{1} + l_{\mathrm{l}}}, \]

it is calculated by the formula

\[ \theta_{\mathrm{m1}} = \theta_{\alpha} + \theta'_{\mathrm{ct}} + \theta_{\Delta} + \theta_{j} + \theta_{z} + \theta_{\mathrm{i}} + \frac{l_{\mathrm{l}}}{l_{1} + l_{\mathrm{l}}}\theta_{\mathrm{lp}} . \tag{31} \]

A comparison of calculated and experimental data for individual temperature drops in rated operation, given in Table 1, indicates their satisfactory agreement.

Table 2 gives a comparison of the results of thermal calculation of motors of the AO2 series, frame sizes 3 and 4, with data from factory tests of these motors, in a quantity of \(N\) units of each type of the basic design.

Table 1

Temperature rises in the AO2-42-4 motor, 5.5 kW, 220/380 V
(experimental data for 8 motors of the same type)

Table 2

Calculated and experimental data on winding heating of AO2 series motors of frame sizes 3 and 4

$\Delta\theta$—the discrepancy between the calculated overheating of the stator winding and its mean experimental value—does not, as a rule, exceed 6% and lies within the limits of the experimental root-mean-square scatter $\Delta\sigma$, which, as can be seen from the table, ranges from 5 to 10%. Satisfactory agreement was also obtained in calculating the heating of experimental motors of the new D and AO4 series.

Thus, the proposed method of thermal calculation has sufficiently high accuracy.

REFERENCES

1. I. M. Postnikov. Design of Electrical Machines. Gostekhizdat, Kiev, 1960.

2. G. G. Schastlivyi. Heating of Enclosed Asynchronous Electric Motors. “Naukova Dumka,” Kiev, 1966.

3. V. A. Zhadan, D. I. Sannikov, R. Ya. Klyain. Study of temperature fields of enclosed ventilated electric motors, Izvestiya TPI, 172, 1967.