Thermal Calculation of Protected DC Machines of the P2 Series
Abstract Generated abstract
This paper develops a thermal calculation method for the armature of protected P2-series DC machines with axial ventilation, axis heights up to 200 mm, and speeds up to 3000 rpm. The method models the armature as a single thermal body and introduces specific internal thermal resistance and an equivalent heat-transfer coefficient to combine heat exchange from the active armature, end windings, and commutator, with empirical coefficients reflecting air-flow conditions. Formulae are derived for thermal resistances, heat-transfer coefficients, air heating, and armature temperature rise, and the procedure is illustrated for P2 machines with 112 and 160 mm axis heights. Comparison with experimental data indicates that the simplified calculation gives temperature-rise estimates with relatively small deviations while reducing computational effort.
Full Text
THERMAL CALCULATION OF PROTECTED DC MACHINES OF THE P2 SERIES
D. I. Sannikov, R. Ya. Klyain
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
This paper presents a method for the thermal calculation of the armature of DC machines with an axis height \(h\) up to 200 mm and a rotational speed up to 3000 rpm, with an axial ventilation system without ventilation ducts.
The proposed method takes into account the principal factors affecting heat exchange in the machine in accordance with generally accepted views on the heat-transfer process in electrical machines, as well as the results of experimental investigations of heat transfer carried out by the authors [3].
To calculate the heating of the armature, the equivalent thermal substitution circuit in Fig. 1 is considered.
Fig. 1. Simplified equivalent thermal circuit of the armature
Basic assumptions:
- The temperature distribution along the length of the armature winding in machines of low power (within 20–30 kW) is practically uniform. In connection with this, the armature is considered as a single body to which, in addition to the winding, the core and collector also belong, and it is characterized by the effective cooling surface
\[ S_{\text{я}} = \pi D l_{\text{э}}, \tag{1} \]
where
\(D\) is the armature diameter,
\(l_{\text{э}}\) is the equivalent armature length.
- The heat released in the pole coils is carried away by the air flow in the interpolar ducts, and heat exchange between the armature and the po-
between the poles is absent. This assumption makes it possible to consider the armature as insulated from the poles.
- The distribution of the cooling-air temperature along the length of the machine obeys a linear law, and allowance for air heating when determining the temperature rise of the armature winding is made by adding the average heating found from the formula:
\[ \Delta \theta_{f\,\mathrm{avg}}=\frac{k_f \sum P_{\mathrm{gr}}}{2 C_p V}, \tag{2} \]
where
\(k_f\) — air-heating coefficient, taking into account that part of the losses is removed, bypassing the ventilation system, to structural parts by heat conduction and radiation, and also that part of the heat from the air is removed to the frame and end shields;
\(P_{\mathrm{gr}}\) — total heating losses, excluding mechanical losses;
\(C_p\) — heat capacity of air;
\(V\) — air flow rate.
- The cooling conditions of the front and rear end parts are identical, and the heat-transfer coefficient \(\alpha_l\) is averaged.
The thermal conductances and resistances of the circuit in Fig. 1 are determined as follows. The thermal conductance from the surface of the active part of the armature to the internal air is
\[ \Lambda_a=\alpha_a \pi D l_a, \tag{3} \]
\(\alpha_a\) — heat-transfer coefficient of the active part of the armature,
\(l_a\) — length of the active part.
The thermal conductance of the surface of both end parts is
\[ \Lambda_l=2\alpha_l D_l f_l', \tag{4} \]
\(\alpha_l\) — average heat-transfer coefficient of the end parts of the armature winding,
\(D_l\) — diameter of the end part,
\(f_l'\) — calculated overhang of the end part.
On the basis of an analysis of P-series machines of frame sizes 1–6, it has been established that for two-pole machines, on average,
\[ f_l'=0.6D, \tag{5} \]
and for four-pole machines
\[ f_l'=0.45D. \tag{6} \]
The conductance of the commutator is
\[ \Lambda_k=1.4\alpha_k \pi D_k l_k, \tag{7} \]
where
\(\alpha_k\) — heat-transfer coefficient of the commutator surface;
\(D_k\) — commutator diameter;
\(l_k\) — commutator length.
The coefficient 1.4 in formula (7) takes into account the increase in the heat-transfer surface of the commutator due to heat removal by the brushes.
The thermal resistance between the copper of the winding and the steel of the core \(R_a\) consists of the thermal resistance of the slot insulation \(R_{\mathrm{ia}}\) and the thermal resistance of the armature teeth \(R_z\):
\[ R_a=R_{\mathrm{ia}}+R_z. \tag{8} \]
The thermal resistance of the slot insulation is determined from the expression [4]
\[ R_{\mathrm{ia}}=\frac{k_p \Delta_i}{z l_a \Pi \lambda_i}, \tag{9} \]
\(\Delta_{\mathrm{i}}\) — thickness of the slot insulation;
\(z\) — number of slots;
\(\Pi\) — cooled perimeter of the slot;
\(\lambda_{\mathrm{i}}\) — coefficient of thermal conductivity of the slot insulation;
\(K_{\mathrm{p}}>1\) — accounts for the increase in thermal resistance due to the insulation of the conductors of the random winding.
The thermal resistance of the armature teeth over \(1/3\) of the height is
\[ R_z=\frac{h_z}{3z l_a b_z \lambda_{\mathrm{Fe}} k_{\mathrm{Fe}}}, \tag{10} \]
where
\(h_z\) — tooth height;
\(b_z\) — tooth width;
\(\lambda_{\mathrm{Fe}}\) — specific thermal conductivity of steel;
\(k_{\mathrm{Fe}}\) — coefficient of filling of the stack with steel.
Let us introduce the concept of the specific internal thermal resistance, i.e., the resistance referred to a unit area of the cooling surface. For the active part
\[ M_a=\pi D l_a R_a=\pi D l_a\left(R_{\mathrm{ia}}+R_z\right). \tag{11} \]
After substituting (9) and (10) into (11) and carrying out a number of transformations, we obtain
\[ M_a=\frac{t_1}{\Pi}\,\frac{\Delta_{\mathrm{i}}k_{\mathrm{p}}}{\lambda_{\mathrm{i}}} +\frac{1}{3\lambda_{\mathrm{Fe}}k_{\mathrm{Fe}}}\,\frac{t_1}{b_z}\,\frac{h_z}{D}\,D, \tag{12} \]
where
\(t_1\) — armature tooth pitch.
The complexes \(\dfrac{\Delta_{\mathrm{i}}k_{\mathrm{p}}}{\lambda_{\mathrm{i}}}\) and \(\dfrac{1}{3\lambda_{\mathrm{Fe}}k_{\mathrm{Fe}}}\) for machines with the same insulation class and grade of steel are approximately constant. The dimensional ratios entering into (12), \(\dfrac{t_1}{\Pi}\), \(\dfrac{t_1}{b_z}\), and \(\dfrac{h_z}{D}\), can, with some approximation, be represented as power functions of \(D\). On the basis of an analysis of the P 1–6 frame-size portion of the series, the following dependence was established:
for insulation class A
\[ M_a=\frac{235}{D^{\frac{1}{2}}}+1.4D^{\frac{1}{2}}\simeq \frac{150}{D^{\frac{1}{2}}}, \tag{13} \]
(\(D\) is expressed in cm);
for any insulation class
\[ M_a=k_{\mathrm{i}}\frac{120}{D^{\frac{1}{2}}}, \tag{14} \]
where for class A \(k_{\mathrm{i}}=1.25\), and for classes B, E, F, H \(k_{\mathrm{i}}=1.0\).
The error of formula (14) in comparison with an exact calculation does not exceed 6%.
Carrying out an analogous analysis for the specific thermal resistances of the end parts and the commutator,
\[ M_{\ell}=\pi D_{\ell} f_{\ell} R_{\ell}, \tag{15} \]
\[ M_k=1.4\pi D_k l_k R_k, \tag{16} \]
on the basis of the method of [4], we obtain, for the portion of the series under consideration, their expression through \(D\) and certain other parameters.
\[ M_l = k_i \left( 1.9 + \frac{1.5}{p^{1/3}} D \right) - k_i^2 \frac{\sqrt{D_p}}{30}, \tag{17} \]
\[ M_k = k_i 4 \frac{D_k l_k}{D} \left( \sqrt{w_s} - \frac{1}{\sqrt{w_s}} \right). \tag{18} \]
Here
\(n\) — rotational speed, rpm;
\(p\) — number of pole pairs;
\(w_s\) — number of turns in a section (when \(w_s\) is variable, its average fractional value is taken).
The formula for determining \(M_l\) was obtained taking into account the influence of band insulation on the end parts on the thermal resistance. The expression for \(M_k\) takes into account the axial thermal resistance of the leads from the sections to the commutator bars and the resistance of the insulation between the leads and the end parts of the winding. Since at \(w_s = 1\) the latter resistance is absent, the thermal resistance between the commutator and the nearest end of the end-winding overhang is equal to zero, as follows from formula (18).
The heat-transfer coefficients in expressions (3), (4), and (7) are calculated from empirical formulas obtained on the basis of generalizing a large number of type tests of machines of the P series, as well as the results of investigations of heat-transfer coefficients on models
\[ \alpha_a = 0.95 \cdot 10^{-3} \left( 1 + 0.6 v_a^{0.9} \right) k_a, \tag{19} \]
\[ \alpha_l = 1.25 \cdot 10^{-3} \left( 1 + 0.85 v_a^{0.85} \right) k_l, \tag{20} \]
\[ \alpha_k = 1.6 \cdot 10^{-3} \left( 1 + v_k^{0.7} \right) k_k. \tag{21} \]
Here
\(v_a\) — peripheral speed of the armature;
\(v_k\) — peripheral speed of the commutator.
The coefficients \(k_a\), \(k_l\), and \(k_k\) take into account the cooling conditions of the armature elements and the nature of the air motion as a function of the axial velocity and the filling factor of the interpolar gap
\[ k_0 = \frac{S_k}{S_{kl}}. \]
Figures 2–4 present the dependences
\[ k = f \left( \frac{v_p}{v} \right), \]
according to which \(k_a\), \(k_l\), and \(k_k\) are selected in formulas (19), (20), and (21).
Here
\[ v_p = \frac{V}{S_k} \]
— the flow velocity of air in the interpolar channel;
\[ v_{pl} = \sqrt{\frac{V^2}{S_k S_{kl}}} \]
— the mean geometric velocity of air in the chamber above the end parts;
\(S_k\) — cross-sectional area of the interpolar channel;
\(S_{kl}\) — cross-sectional area of the chamber above the end parts;
\[ v_{pk} = \frac{V}{S_{kl}} \]
— the air velocity in the chamber above the commutator.
Since the armature is considered by us as a single body, it is very convenient for calculation to introduce the concept of an equivalent heat-transfer coefficient, which takes into account both the cooling conditions of the surface and the thermal resistance of the common and turn insulation.
Fig. 2. Dependence of the coefficient \(k_a\) for the active part of the armature on air velocity at
\(n = 1500\) rpm and \(n = 3000\) rpm.
Fig. 3. Dependence of the coefficient \(k_\ell\) for the frontal parts of the armature on air velocity at
\(n = 1500\) rpm
and \(n = 3064\) rpm.
Fig. 4. Dependence of the coefficient \(K_k\) for the commutator at
\(n=1500\) rpm
and \(n=3000\) rpm
\(Q_{\mathrm{n}}\) is determined from the expression
\[ \alpha_{\mathrm{eq}} = \left( \frac{\alpha_a \zeta_a}{1+M_a \alpha_a} + \frac{\alpha_l \zeta_l}{1+M_l \alpha_l} + \frac{\alpha_k \zeta_k}{1+M_k \alpha_k} \right) k_p . \tag{22} \]
Here
\[ \zeta_a=\frac{l_a}{l_e}; \qquad \zeta_l=\frac{1.9 f_l'}{l_e}; \qquad \zeta_k=\frac{1.4 l_k}{l_e}\frac{D_k}{D} \tag{23} \]
— fractional values of the lengths of the armature sections,
\[ l_e = l_a + 1.9 f_l' + 1.4 l_k \frac{D_k}{D}. \tag{24} \]
The equivalent armature length \(l_e\) is the sum of the lengths of the active part, the end parts, and the commutator, reduced to the diameter of the active part of the armature. Since in the machines considered \(D_l/D\) is on average 0.95, the reduced length of the end overhang is taken with the coefficient 1.9.
\[ k_p = \frac{1}{ 1 - \frac{M_a \alpha_a}{1+M_a \alpha_a} \frac{P_{\mathrm{st}}+0.5P_{\mathrm{add}}}{P_{\mathrm{a}}} - \frac{M_k \alpha_k}{1+M_k \alpha_k} \frac{P_k}{P_l} } \tag{25} \]
— coefficient of the loss ratio;
\(P_{\mathrm{st}}\) — iron losses;
\(P_{\mathrm{add}}\) — additional losses;
\(P_k\) — total losses in the commutator;
\(P_ya = P_{mya} + P_{st} + P_k + 0.5P_{dob}\) — total losses in the armature;
\(P_{mya}\) — losses in the armature winding.
The temperature rise of the armature above the ambient air is determined by the formula
\[ \theta_{ya}=\frac{P_{ya}}{k_p S_{ya}\alpha_{eya}}+\Delta\theta_{f\mathrm{avg}} . \tag{26} \]
The average heating of the air \(\Delta\theta_{f\mathrm{avg}}\) is found from (2), where \(k_f=0.9\),
\[ P_{gr}=P_{ya}+P_p, \tag{27} \]
\(P_p\) — total losses in the pole coils.
The tables give a detailed calculation of the armature of P2 machines with axis height of rotation \(h=112\) and \(160\) mm.
Table 1 gives the initial data; Table 2 gives the intermediate and final calculation results and comparison with experiment.
Main Conclusions
- In the proposed method for thermal calculation of the armature, formulas are used for calculating the heat-transfer coefficients of the armature elements, taking into account the nature of the air-flow motion.
Table 1
| No. | Quantity | Unit | P2 = 160, \(n=1500\) | P2 = 160, \(n=3000\) | P2 = 112, \(n=1500\) | P2 = 112, \(n=3000\) |
|---|---|---|---|---|---|---|
| 1. | \(U_{ya}\) | V | 228 | 440 | 220 | 440 |
| 2. | \(I_{ya}\) | A | 60 | 65 | 13.7 | 16.5 |
| 3. | \(P_{mya}\) | W | 525 | 684 | 250 | 372 |
| 4. | \(P_{st}\) | W | 130 | 420 | 39 | 182 |
| 5. | \(P_{dob}\) | W | 137 | 286 | 32.3 | 72.6 |
| 6. | \(P_p\) | W | 717 | 855 | 390 | 474 |
| 7. | \(P_k\) | W | 154 | 223 | 47.2 | 72.5 |
| 8. | \(P_{ya}\) | W | 947 | 1470 | 352 | 663 |
| 9. | \(P_D + P_p\) | W | 1664 | 2325 | 742 | 1137 |
| 10. | \(D\) | cm | 16.2 | 16.2 | 10.6 | 10.6 |
| 11. | \(D_k\) | cm | 12.5 | 12.5 | 8.0 | 8.0 |
| 12. | \(l_a\) | cm | 17 | 17 | 15 | 15 |
| 13. | \(l_k\) | cm | 7 | 7 | 3.7 | 3.7 |
| 14. | \(f_p\) | cm | 7 | 7 | 6 | 6 |
| 15. | \(l_e\) | cm | 37.7 | 37.7 | 30.3 | 30.3 |
| 16. | \(\zeta_a\) | 0.45 | 0.45 | 0.5 | 0.5 | |
| 17. | \(\zeta_p\) | 0.375 | 0.375 | 0.35 | 0.35 | |
| 18. | \(\zeta_k\) | 0.20 | 0.20 | 0.125 | 0.125 | |
| 19. | \(S_k\) | \(\mathrm{m^2}\) | \(64.5\cdot10^{-4}\) | \(64.5\cdot10^{-4}\) | \(63\cdot10^{-4}\) | \(63\cdot10^{-4}\) |
| 20. | \(S_{kl}\) | \(\mathrm{m^2}\) | \(368\cdot10^{-4}\) | \(368\cdot10^{-4}\) | \(180\cdot10^{-4}\) | \(180\cdot10^{-4}\) |
| 21. | \(2p\) | 4 | 4 | 4 | 4 |
- The concepts of “specific internal thermal resistance” and “equivalent heat-transfer coefficient” have been introduced; these made it possible to substantially simplify the calculation of thermal resistances and to reduce the amount of computational work practically without loss of accuracy.
Table 2
| No. | Designation | P2-160, \(n=1500\) | P2-160, \(n=3000\) | P2-112, \(n=1500\) | P2-112, \(n=3000\) |
|---|---|---|---|---|---|
| 1. | \(V\) | 0,047 | 0,082 | 0,018 | 0,036 |
| 2. | \(v_a\) | 12,7 | 25,7 | 8,3 | 16,6 |
| 3. | \(M_a\) | 20 | 20 | 25 | 25 |
| 4. | \(M_l\) | 20 | 20 | 18 | 18 |
| 5. | \(M_k\) | 13 | 13 | 12,5 | 12,5 |
| 6. | \(v_p\) | 6,3 | 12,7 | 2,86 | 5,72 |
| 7. | \(v_{p1}v_a\) | 0,496 | 0,5 | 0,345 | 0,345 |
| 8. | \(k_a\) | 0,875 | 1,01 | 0,846 | 0,9 |
| 9. | \(\alpha_a \cdot 10^3\) | 5,76 | 11,5 | 4,05 | 7,33 |
| 10. | \(\alpha_a M_a\) | 0,115 | 0,23 | 0,101 | 0,183 |
| 11. | \(\alpha_{ae}\) | 5,17 | 9,35 | 3,68 | 6,19 |
| 12. | \(\alpha_{ae}\xi_a\) | 2,32 | 4,21 | 1,84 | 3,1 |
| 13. | \(v_{pl}\) | 2,64 | 5,33 | 1,69 | 3,38 |
| 14. | \(v_{pl}/v_a\) | 0,208 | 0,21 | 0,204 | 0,204 |
| 15. | \(k_l\) | 1,02 | 1,135 | 0,975 | 1,12 |
| 16. | \(\alpha_l \cdot 10^3\) | 10,7 | 20,3 | 7,5 | 14,45 |
| 17. | \(\alpha_l M_l\) | 0,214 | 0,406 | 0,135 | 0,26 |
| 18. | \(\alpha_{le}\) | 8,81 | 14,4 | 6,61 | 11,5 |
| 19. | \(\alpha_{le}\xi_l\) | 3,09 | 5,04 | 2,48 | 4,31 |
| 20. | \(v_{pk}\) | 1,1 | 2,23 | 1,0 | 2,0 |
| 21. | \(v_{pk}/v_k\) | 0,112 | 0,113 | 0,159 | 0,159 |
| 22. | \(k_k\) | 1,09 | 1,06 | 1,05 | 1,05 |
| 23. | \(\alpha_k \cdot 10^3\) | 10,35 | 15,4 | 7,74 | 11,55 |
| 24. | \(\alpha_k M_k\) | 0,1345 | 0,2 | 0,0967 | 0,1445 |
| 25. | \(\alpha_{ke}\) | 9,12 | 12,8 | 7,06 | 10,1 |
| 26. | \(\alpha_{ke}\xi_k\) | 1,825 | 2,56 | 0,882 | 1,26 |
| 27. | \(\sum \chi_i \xi_i\) | 7,235 | 11,81 | 5,2 | 8,67 |
| 28. | \(k_p\) | 1,042 | 1,11 | 1,027 | 1,07 |
| 29. | \(\alpha_{eya}\) | 7,55 | 13,1 | 5,35 | 9,27 |
| 30. | \(P_{ya}/S_{ya}\cdot\alpha_{eya}\) | 65,3 | 58,5 | 65,1 | 70,8 |
| 31. | \(\Delta \Theta_{fcp}\) | 18,6 | 12,9 | 18,8 | 14,3 |
| 32. | \(\Theta_{ya}\) | 83,9 | 71,4 | 83,7 | 85,1 |
| 33. | \(\Theta_{ya,opyt}\) | 79/81,5 | 79,5/80,3 | 80/77 | 79,5 |
| 34. | \((\Theta_{ya}-\Theta_{ya,op})\cdot100/\Theta_{ya,op}\) | 6/3 | -10/-11 | 5/9 | 7 |
Note:
-
Since slot insulation with a higher heat-transfer coefficient (NEM composition) was used in the motor mock-up samples under consideration, the formulas for calculating \(M_a\), \(M_l\), and \(M_k\) are multiplied by the coefficient \(2/3\).
-
The experimental value \(\Theta_{ya,op}\) in item 33 is given for two machines.
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. Methodology for the thermal calculation of protected high-speed DC motors. — Elektromekhanika, 1964, No. 5.
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R. Ya. Klyain, D. I. Sannikov. Investigation of heat-transfer coefficients on mock-ups of low-power DC machines. Proceedings of TPI, vol. 211, 1970.
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D. I. Sannikov, R. Ya. Klyain. Thermal calculation of the armature of low-power DC machines. Proceedings of TPI, vol. 212, 1971.