Thermal calculation of the armature of low-power DC machines
Abstract Generated abstract
A thermal calculation method is proposed for protected self-ventilated low-power DC machines with armature diameters up to 200 mm and speeds up to 4000 rpm. The method simplifies the armature winding as a uniformly heated body while accounting for different heat-transfer coefficients, insulation and steel thermal resistances, banding of end windings, internal air heating, and thermal coupling between the winding and commutator through section leads. Formulae are developed for the main thermal resistances, conductances, reduced losses, and winding temperature rise. Comparison with experimental data for several series motors shows discrepancies within about 6 percent, which the authors regard as comparable with ordinary experimental scatter.
Full Text
THERMAL CALCULATION OF THE ARMATURE OF LOW-POWER DC MACHINES
D. I. Sannikov, R. Ya. Klyayn
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
The proposed method has been developed for protected self-ventilated DC machines having an armature diameter \(D_a \le 200\) mm and a rotational speed of up to 4000 rpm. The armatures of these machines are usually made without ventilation ducts and have a winding of soft sections, which makes cooling of the end parts difficult. As experiments show, this circumstance, as well as the considerable axial thermal conductivity of the winding, leads to an equalization of the temperature distribution along the length of the armature, which makes it possible to simplify the thermal-calculation method substantially by treating the winding as a single uniformly heated body.
This method takes into account all the most important factors in accordance with the views currently accepted:
the difference between the heat-transfer coefficients on the surfaces of the core and the end parts of the armature;
the thermal resistance of the insulation, both slot insulation and turn insulation, and of the core steel;
the influence of bands on the end parts;
heating of the air in the machine;
the connection of the winding with the commutator through the section leads.
Allowing for the last factor as applied to soft multi-turn sections is one of the special features of this method, as is the following: since the lower layer of the end parts of the winding is not cooled, the heat flow from it spreads along the winding conductors and passes into the upper layer, creating an additional internal temperature drop in the end parts, which is taken into account by a special thermal resistance.
The main influence on the accuracy of the calculation is exerted by the correct choice of thermophysical constants. On the basis of experimental data, the following expressions may be recommended for the heat-transfer coefficients of the active part of the armature, the end parts, and the commutator:
\[ \alpha_a = 0.77 v_a^{0.85}\cdot 10^{-3},\ \frac{\text{W}}{\text{cm}^2\cdot{}^\circ\text{C}}, \tag{1} \]
\[ \alpha_l = 1.3\cdot 10^{-3}\left(1 + 0.85v_a^{0.85}\right), \tag{2} \]
\[ \alpha_k = 1.7\cdot 10^{-3}\left(1 + v_k^{0.7}\right). \tag{3} \]
Here \(v_a\) and \(v_k\) are the peripheral velocities of the armature and commutator, m/s. The thermal-conductivity coefficients of insulation are: for class A, \(\lambda_i = 0.001\ \text{W}/(\text{cm}\cdot{}^\circ\text{C})\); for the other classes, \(\lambda_i = 0.0016\).
Equivalent thermal-conductivity coefficient of the random winding
\[ \lambda_{\mathrm{e}}=\frac{2.5\lambda_{\mathrm{i}}}{\sqrt{k\frac{d'}{d}-1}} . \tag{4} \]
Here \(d'\) and \(d\) are the diameters of the insulated and bare conductor; for the slot portion \(k=1.3\), and for the end-winding portion \(k=1.1\).
The armature dimensions required for the calculation are shown in Fig. 1 and are taken in centimeters.
The thermal resistance of the insulation of the slot portion of the winding \((\mathrm{deg}/\mathrm{W})\) is
\[ R_{\mathrm{ia}}=\frac{1}{z\,l_a} \left[ \frac{\Delta_{\mathrm{i}}}{\Pi\lambda_{\mathrm{i}}} + \frac{1}{12\lambda_{\mathrm{e}}\left(\nu+\frac{1}{3\nu}\right)} \right], \tag{5} \]
Fig. 1
where
\(z\) — number of armature teeth,
\(\Delta_{\mathrm{i}}\) — thickness of the slot insulation,
\[ \Pi=2h_0+\pi r_2+\frac{2}{3}r_1 \tag{6} \]
is the cooled perimeter of the slot,
\[ \nu=\frac{h_0-h_{\mathrm{cl}}}{r_1+r_2}+\frac{\pi}{4} \tag{7} \]
(the ratio of the mean slot height to the mean width, without allowance for the wedge).
The resistance of the armature teeth over \(1/3\) of their height is
\[ R_z=\frac{h_0+0.5r_1+r_2}{3z\,l_a\,b_z\,\lambda_{\mathrm{Fe}}\,k_{\mathrm{Fe}}}, \tag{8} \]
\(\lambda_{\mathrm{Fe}}=0.41\ \mathrm{W}/\mathrm{cm\,deg}\) — thermal conductivity of steel,
\(k_{\mathrm{Fe}}\) — steel stacking factor.
The resistance between the surface of the active part of the armature and the air is
\[ R_\alpha=\frac{1}{\pi D_a l'_a \alpha_a}. \tag{9} \]
The coefficient for reducing the iron losses to the winding is
\[ k_F=\frac{1}{1+\dfrac{R_{\mathrm{ia}}}{R_\alpha+R_z}} . \tag{10} \]
Calculated thickness of the end-winding parts (without allowance for spacers)
\[ h_l = 0.24\,\frac{(d')^2 N_p z l_l}{f_l D_a}. \tag{11} \]
Here \(N_p\) is the number of elementary conductors in a slot,
\(l_l\) is the length of the end-winding part.
Calculated projection of the end-winding parts
\[ f'_l = f_l + h_l. \]
Cooling surface
\[ S_l = 0.95\pi D_a f'_l. \]
Total cross-sectional area of the conductors
\[ S_m = zN_p\,\frac{\pi}{4}\,(d')^2. \tag{14} \]
Thermal resistance of the turn insulation (per one side of the armature)
\[ R_{il}=\frac{h_l}{12\lambda_e S_l}. \tag{15} \]
Axial thermal resistance of the end-winding parts
\[ R_{ml}=\frac{2l_l}{S_m\lambda_{cu}}. \tag{16} \]
Here \(\lambda_{cu}=3.83\) is the thermal conductivity of copper.
Taking into account that the maximum temperature difference between the upper and lower layers is observed at a distance of \(2/3\) from the end of the projection, and that the excess of the average temperature of the end-winding part over the temperature of the upper layer is \(\Delta\Theta_m/3\), and assuming that 90% of the losses are transferred from the lower layer to the upper layer, we obtain that the corresponding thermal resistance is \(R_{ml}/30\).
The thermal resistance between the surface of the end-winding parts and the air, allowing for the insulation under the bandage, which has thickness
\[ R_{la}= \frac{1}{ \alpha_l S_l\frac{f'_l-l_b}{f'_l} + S_l\frac{l_b}{f'_l} \frac{1}{\frac{1}{\alpha_l}+\frac{\Delta_b}{\lambda_i}} } = \tag{18} \]
\[ = \frac{1}{ \alpha_l S_l \left( 1-\frac{l_b}{f'_l}\, \frac{1}{1+\frac{\lambda_i}{\Delta_b\alpha_l}} \right) }. \tag{17} \]
Thermal conductance from the commutator to the air
\[ G_k=\pi D_k l_k \alpha_k. \]
The thermal resistance between the commutator and the winding is determined from the following considerations. The section leads laid along the surface of the end-winding parts have a thermal connection with them characterized by the resistance \(R_{il}\); the axial resistance of the leads has the value \(0.25\,R_{ml}w_s\). The temperature difference between the leads and the winding decreases from the commutator toward the armature according to an exponential law with attenuation coefficient
\[ \beta=\frac{2}{l_l}\sqrt{\frac{0.25\,R_{ml}w_s}{R_{il}}}. \]
The resistance equal to the ratio of the temperature difference between the commutator and the end of the overhang of the end portions to the heat flow commutator--winding is equal to \(R'_{kl}=R_{il}\beta\).
If the mean temperature of the winding is to be determined taking into account the fact that the leads also belong to the number of conductors of the end portions, then the resistance obtained must be multiplied by the ratio \((w_s-1)/w_s\).
Finally, in view of the fact that there is a section of the leads directly adjoining the commutator and not in contact with the end portions, we increase the resistance by 20%. Thus, finally we have
\[ R_{kl}=1.2\frac{w_s-1}{w_s}R_{il}\beta =0.6(w_s-1)\sqrt{\frac{R_{ml}R_{il}}{w_s}} . \tag{19} \]
The coupling coefficient of the commutator with the winding is
\[ k_{kl}=\frac{1}{1+G_kR_{kl}} . \tag{20} \]
The resulting thermal conductance from the armature winding to the air is
\[ G_ya=\frac{1}{R_a+R_z+R_{na}} +\frac{2}{R_{la}+\frac{1}{30}R_{ml}+R_{il}} +k_{kl}G_k . \tag{21} \]
The reduced armature losses are determined from the formula
\[ P_ya=P_{mya}+k_F P_{st}+0.5P_{dob}+k_{kl}0.7P_k, \tag{22} \]
where the losses in the winding \(P_{mya}\), in the steel \(P_{st}\), the additional losses \(P_{dob}\), and the commutator losses \(P_k\) are taken into account.
The heating of the air in the machine is
\[ \Delta\theta_f=\frac{0.9P_{gr}}{1100V_p}. \tag{23} \]
Here \(P_{gr}\) is the total heating loss,
\(V_p\) is the cooling-air flow rate.
The temperature rise of the armature winding is
\[ \theta_ya=\frac{P_ya}{G_ya}+\frac{\Delta\theta_f}{2}. \tag{24} \]
As an example, the table gives the results of the thermal calculation of several motors and compares them with experimental data.
Conclusions
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The proposed method more accurately takes into account the temperature differences between the internal parts of the winding and the cooled surfaces, as well as the connection with the commutator.
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The error of the calculation does not exceed the usual scatter of experimental data.
Table
Calculated and experimental data on heating of the armature of series П motors
| Motor type | П12 | П32 | П41 | П22* | П52* | П62* |
|---|---|---|---|---|---|---|
| Power, kW | 0,45 | 2,2 | 3,2 | 2,9 | 12,4 | 37,0 |
| Rotational speed, rpm | 1500 | 1500 | 1500 | 3870 | 2750 | 3810 |
| Voltage, V | 220 | 220 | 220 | 320 | 220 | 320 |
| Winding temperature rise, deg, calculation | 56,0 | 57,3 | 91,9 | 52,0 | 83,1 | 94,1 |
| Winding temperature rise, deg, experiment | 53,0 | 61,0 | 88,0 | 51,5 | 82,0 | 89,0 |
| Discrepancies, deg | +3,0 | −3,3 | +3,9 | +0,5 | +1,1 | +5,1 |
| Discrepancies, % | +5,6 | −5,4 | +4,4 | +1,0 | +1,3 | +5,6 |
Note. * — modification.
REFERENCES
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Ya. S. Gurin and M. N. Kurochkin. Design of DC Machines. Gosenergoizdat, 1961.
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A. I. Borisenko, A. I. Yakovlev. Method for the thermal calculation of protected high-speed DC motors. Elektromekhanika, 1964, No. 5.