ON THE THERMAL CALCULATION OF THE ARMATURE OF ENCLOSED NON-VENTILATED DC MACHINES

M. N. Ulyanitskii, V. V. Salomatov

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

In enclosed DC machines the maximum thermal load is usually borne by the armature winding. This paper presents a method for calculating the temperature field along the length of the armature winding, taking into account the thermal connection with the commutator and the removal of heat along the shaft and through the slot wedge. The temperature in the cross section of the slot and in the radial direction of the core is assumed to be constant; heat transfer in the radial direction of the core and heat emission from its end surface are not taken into account.

The calculation is carried out for one tooth pitch. The winding is considered as a rod having seven sections (Fig. 1): the upper and lower leads to the commutator, the slot portion, the parts of the end connections on the drive side and the commutator side located under the insulation bandage, and the end portions of the end connections on the drive side and the commutator side.

Fig. 1. Calculation diagram of the armature winding.

In the calculation it is assumed that, from the end portions of the end connections on the commutator side, which are covered from above by the section leads, and from the lower winding leads to the commutator, heat is removed only along the conductors.

In accordance with the heat-balance equations [4], the equations of the temperature-distribution curves along the length of the individual sections of the armature winding have the following form:

for the end portion of the end connections on the commutator side and the lower leads,

\[ \vartheta = -\frac{B}{2}x^{2} + T_{1}x + T_{2}. \tag{1} \]

For the remaining parts of the winding, taking

\[ \frac{A'}{m^{2}} = A, \]

\[ V = C_1 \exp(mx) + C_2 \exp(-mx) \]

The values of the integration constants are determined under the following boundary conditions:

1) for the slotted part of the winding

\[ x = -\frac{l_{\mathrm{p}}}{2}, \quad \vartheta_{\mathrm{p}}=\vartheta_1, \qquad x = \frac{l_{\mathrm{p}}}{2}, \quad \vartheta_{\mathrm{p}}=\vartheta_2, \]

\[ C_{1(2)} = \frac{ \left(\vartheta_{2(1)} - A_1\right)\exp m_1 \frac{l_{\mathrm{p}}}{2} - \left(\vartheta_{1(2)} - A_1\right)\exp\left(-m_1 \frac{l_{\mathrm{p}}}{2}\right) }{ 2 \operatorname{sh} m_1 l_{\mathrm{p}} }, \]

\[ A_1'=\frac{k_c q_c+q_m}{\lambda_m f_{\mathrm{p}}}, \qquad k_c=\frac{1}{1+\dfrac{(a b_k+\Lambda_{\mathrm{v}})\Delta_{\mathrm{p}}}{\lambda_{\mathrm{p}}\Pi}}, \qquad \Lambda_{\mathrm{v}}=\frac{\alpha_{\mathrm{v}}\lambda_{\mathrm{v}} l_{\mathrm{v}} \pi D_{\mathrm{v}}}{z(\alpha_{\mathrm{v}}\delta_{\mathrm{v}}l_{\mathrm{v}}+\lambda_{\mathrm{v}}l_{\mathrm{p}})}, \]

\[ m_1^2=\frac{\Lambda}{\lambda_m f_{\mathrm{p}}}, \qquad \Lambda=\Lambda_{\mathrm{Fe}}+\Lambda_k, \qquad \Lambda_{\mathrm{Fe}}=k_c(a b_k+\Lambda_{\mathrm{v}}), \qquad \Lambda_k=\frac{\alpha b_{\mathrm{sh}}\lambda_k}{a\Delta_k+\lambda_k}; \]

2) for the section of the end connections on the drive side under the band insulation

\[ x=\frac{l_{\mathrm{p}}}{2}, \quad \vartheta_l=\vartheta_1, \qquad x=\frac{l_{\mathrm{p}}}{2}+l_i, \quad \vartheta_l=\vartheta_4, \]

\[ C_1= \frac{ \vartheta_4-\vartheta_2 \exp(-m_2 l_i)-A_2\left[1-\exp(-m_2 l_i)\right] }{ 2 \exp m_2 \frac{l_{\mathrm{p}}}{2}\operatorname{sh} m_2 l_i }, \qquad m_2^2=\frac{\Lambda_l}{\lambda_m f_{\mathrm{p}}}, \]

\[ C_2= \frac{ \vartheta_2 \exp m_2 l_i-\vartheta_4+A_2\left(1-\exp m_2 l_i\right) }{ 2 \exp\left(-m_2\frac{l_{\mathrm{p}}}{2}\right)\operatorname{sh} m_2 l_i }, \qquad A_2'=\frac{q_m}{\lambda_m f_{\mathrm{p}}}, \]

\[ \Lambda_l=\frac{b_l \alpha \lambda_l}{a\Delta_l+\lambda_l}; \]

3) for the section of the end connections on the commutator side under the band insulation

\[ x=-\frac{l_{\mathrm{p}}}{2}, \quad \vartheta_{lk}=\vartheta_2, \qquad x=-\left(\frac{l_{\mathrm{p}}}{2}+l_i\right), \quad \vartheta_{lk}=\vartheta_3, \]

\[ C_1= \frac{ \vartheta_1 \exp m_3 l_i-\vartheta_3+A_3\left(1-\exp m_3 l_i\right) }{ 2 \exp\left(-m_3\frac{l_{\mathrm{p}}}{2}\right)\operatorname{sh} m_3 l_i }, \]

\[ C_2= \frac{ \vartheta_3-\vartheta_1 \exp(-m_3 l_i)-A_3\left[1-\exp(-m_3 l_i)\right] }{ 2 \exp m_3 \frac{l_{\mathrm{p}}}{2}\operatorname{sh} m_3 l_i }, \]

\[ m_3^2=\frac{\Lambda_{lk}}{\lambda_m f_l}, \qquad \Lambda_{lk}=\frac{\alpha\lambda_l b_{lk}}{\alpha\Delta_l+\lambda_l}, \qquad A_3'=A_2'; \]

4) for the end face part of the end connections on the drive side

\[ x=\frac{l_{\mathrm{p}}}{2}+l_i, \quad \vartheta_T=\vartheta_4, \qquad x=\frac{1}{2}, \quad \frac{d\vartheta_T}{dx}=0, \]

\[ C_{1(2)}= \frac{\vartheta_4-A_4}{2\operatorname{ch} m_4 l_T} \exp\left[(\mp)m_4\frac{l}{2}\right], \qquad m_4^2=\frac{\alpha b_T}{\lambda_m f_{\mathrm{p}}}, \qquad A_4'=A_2'. \]

5) for the upper leads of the winding to the commutator

\[ x=-\frac{l_{\mathrm{p}}}{2}\quad \vartheta_{\mathrm{v}}=\vartheta_1,\qquad x=-\left(\frac{l_{\mathrm{p}}}{2}+l_{\mathrm{c}}\right)\quad \vartheta_{\mathrm{v}}=\vartheta_{\mathrm{k}}, \]

\[ C_1= \frac{\vartheta_1 \exp m_5 l_{\mathrm{c}}-\vartheta_{\mathrm{k}}+A_5(1-\exp m_5 l_{\mathrm{c}})} {2\exp\left(-m_5 \frac{l_{\mathrm{p}}}{2}\right)\operatorname{sh} m_5 l_{\mathrm{c}}}, \]

\[ C_2= \frac{\vartheta_{\mathrm{k}}-\vartheta_1\exp(-m_5 l_{\mathrm{c}})-A_5[1-\exp(-m_5 l_{\mathrm{c}})]} {2\exp m_5 \frac{l_{\mathrm{p}}}{2}\operatorname{sh} m_5 l_{\mathrm{c}}}, \]

\[ m_5^2=\frac{\Lambda_{\mathrm{c}}}{\lambda_m f_{\mathrm{c}}},\qquad \Lambda_{\mathrm{c}}=\frac{a\lambda_{\mathrm{c}} b_{\mathrm{c}}}{a\Delta_{\mathrm{c}}+\lambda_{\mathrm{c}}},\qquad A_5'=A_2'; \]

6) for the lower leads to the commutator

\[ x=-\left(\frac{l_{\mathrm{p}}}{2}+l_{\mathrm{c}}\right)\quad \vartheta_{\mathrm{n}}=\vartheta_{\mathrm{k}},\qquad x=-\frac{l_{\mathrm{p}}}{2}\quad \vartheta_{\mathrm{n}}=\vartheta_1, \]

\[ T_1=\vartheta_1\frac{1}{l_{\mathrm{c}}}-\vartheta_{\mathrm{k}}\frac{1}{l_{\mathrm{c}}} -\frac{B_1}{2}(l_{\mathrm{p}}+l_{\mathrm{c}}), \]

\[ T_2=\vartheta_1\left(1+\frac{l_{\mathrm{p}}}{2l_{\mathrm{c}}}\right) -\vartheta_{\mathrm{k}}\frac{l_{\mathrm{p}}}{2l_{\mathrm{c}}} -\frac{B_1}{2}\left(\frac{l_{\mathrm{p}}}{2}\right)^2 \left(1+\frac{2l_{\mathrm{c}}}{l_{\mathrm{p}}}\right), \qquad B_1=A_2'; \]

7) for the end part of the end connections on the commutator side

\[ x=-\left(\frac{l_{\mathrm{p}}}{2}+l_{\mathrm{u}}\right)\quad \vartheta_{\mathrm{tk}}=\vartheta_3,\qquad x=-\frac{1}{2}\quad \frac{d\vartheta_{\mathrm{tk}}}{dx}=0, \]

\[ T_1=-B_2\frac{1}{2},\qquad T_2=\vartheta_3-\frac{B_2}{2}\left(\frac{l_{\mathrm{p}}}{2}+l_{\mathrm{u}}\right) \left(\frac{1}{2}+l_{\mathrm{t}}\right), \]

\[ B_2=A_2'. \]

The values \(v_1, v_2, v_3, v_4, v_{\mathrm{k}}\) are determined from the condition of continuity of the heat fluxes in the winding sections

\[ x=\pm\left(\frac{l_{\mathrm{p}}}{2}+l_{\mathrm{p}}\right),\qquad x=\pm\frac{l_{\mathrm{p}}}{2} \]

and from the equation of the heat balance of the commutator.

\[ \vartheta_{\mathrm{k}}\alpha_{\mathrm{k}}S_{\mathrm{k}} =q_{\mathrm{k}}+\lambda_m fc\left( \frac{d\vartheta_{\mathrm{v}}}{dx} +\frac{d\vartheta_{\mathrm{n}}}{dx} \right)_{x=-\left(\frac{l_{\mathrm{p}}}{2}+l_{\mathrm{c}}\right)}. \]

\[ \vartheta_1= \frac{ \left(\frac{a_1}{f_{\mathrm{p}}}P_6P_7+P_5P_9\right)(P_1P_3-1) +\left(P_1P_4+P_2\right)\frac{P_5}{m_2} }{ \left(P_5P_8-\frac{a_1}{f_{\mathrm{p}}}P_6^2\right)(P_1P_3-1) -\frac{a_3}{a_1}P_5P_1 }, \]

\[ \vartheta_2= \frac{ m_2\vartheta_1\frac{a_2}{a_1}P_1+(P_1P_4+P_2) }{ m_2(P_1P_3-1) }, \]

\[ \vartheta_3= \frac{ m_3\vartheta_1+(a_5+B_1l_{\mathrm{r}})\operatorname{sh} m_3l_{\mathrm{u}} }{ m_3\operatorname{ch} m_3l_{\mathrm{u}} }, \]

\[ \vartheta_4= \frac{ \vartheta_2+\frac{P_2}{m_2} }{ P_1 }, \qquad \vartheta_{\mathrm{k}}=\lambda_m\frac{\vartheta_1P_6+P_7}{P_5}. \]

Here

\[ a_1=\frac{\operatorname{sh} m_1 l_p}{m_1},\qquad a_2=\frac{\operatorname{sh} m_2 l_i}{m_2},\qquad a_3=\frac{\operatorname{sh} m_5 l_c}{m_5}, \]

\[ a_4=\frac{A_1}{a_1}\left(\operatorname{ch} m_1 l_p-1\right),\qquad a_5=\frac{A_2}{a_2}\left(\operatorname{ch} m_2 l_i-1\right), \]

\[ a_6=\frac{A_5}{a_3}\left(\operatorname{ch} m_5 l_c-1\right),\qquad P_1=\operatorname{ch} m_2 l_i+a_2 m_4 \operatorname{th} m_4 l_t, \]

\[ P_2=\left(a_5+m_4 A_4 \operatorname{th} m_4 l_t\right)\operatorname{sh} m_2 l_i,\qquad P_3=\operatorname{ch} m_2 l_i+\frac{a_2}{a_1}\operatorname{ch} m_1 l_p, \]

\[ P_4=(a_1+a_2)\operatorname{sh} m_2 l_i,\qquad P_5=\left(\frac{a_k S_k}{\lambda_m}+\frac{f_c}{\operatorname{th} m_5 l_c}+\frac{f_c}{l_c}\right)\lambda_m^2, \]

\[ P_6=\left(\frac{1}{a^3}+\frac{1}{l_c}\right)\lambda_m f_c,\qquad P_7=(a_6+0.5B_1l_c)\lambda_m f_c+q_k, \]

\[ P_8=\left(\frac{f_p}{a_1}\operatorname{ch}m_1l_p+\frac{f_c}{a_3}\operatorname{ch}m_5l_c+f_l m_3 \operatorname{th}m_3l_i+\frac{f_c}{l_c}\right)\frac{a_1}{f_p}, \]

\[ P_9=\left(f_p a_4+f_l a_5+f_c a_6+0.5 f_c B_1+\frac{a_5+B_1l_t}{\operatorname{ch}m_3l_i}f_l\right)\frac{a_1}{f_p}. \]

Verification of the calculation method considered showed that the calculated temperature values agree sufficiently closely with the experimental data (Fig. 2). In the calculations, the mean temperature of the frame was taken as the ambient temperature [5].

Fig. 2. Temperature distribution along the length of the armature winding of an enclosed non-ventilated motor of type PB42.
\((2f_c/f_p=0.17,\ \vartheta_k=45^\circ\mathrm{C})\)

● — experiment. Calculated curves:
1 — basic variant; 2 — \(\vartheta_k=0.8\,\vartheta_{a,\mathrm{avg}},\ 2f_c/f_p=0.3\); 3 — \(\vartheta_k=\vartheta_{a,\mathrm{avg}}\);
4 — without taking into account heat removal along the shaft;
5 — without taking into account heat removal through the slot wedge;
6 — without taking into account thermal coupling with the commutator, heat removal along the shaft and through the slot wedge.

To determine the influence on the character of the temperature distribution of the thermal coupling of the armature with the commutator, heat removal along the shaft and through the slot wedge, a number of calculation variants were carried out.

Heat exchange between the commutator and the armature winding is determined mainly by the difference in their temperatures, and also by the ratio of the cross-sectional areas of the sections and the leads. The length of the latter in low-power machines varies within comparatively small limits and therefore plays a lesser role.

The strongest thermal connection with the commutator, other conditions being equal, will be possessed by windings with rigid sections and by random-wound windings of low-voltage machines, in which the ratio of the cross-sectional areas of the sections reaches a value of 0.5–1.0.

According to comparative calculations, for random-wound windings with \(2f_c/f_p < 0.3\) and \(\vartheta_k = (0.8 \div 1.2)\vartheta_{a\,\mathrm{av}}\), the commutator affects mainly only the average temperature level. Neglecting heat exchange with the commutator in these cases leads to a change in the difference between the temperature values at individual points along the length of the armature winding by approximately 1–2%.

In calculating heat removal through the shaft, the main difficulty is caused by determining the value of the thermal resistance of the contact zone between the shaft and the armature stack. Contact between solid bodies is always discrete in character. Moreover, even under comparatively strong compression, the area of actual contact constitutes an insignificant fraction of the area of contact determined by the geometrical dimensions of the bodies in contact [1].

The magnitude of the thermal resistance of the shaft–stack contact zone in the general case depends on the finish of the mating surfaces, the accuracy class and character of the fit, the hardness of the materials, and also on possible technological deviations during manufacture of the machine (misalignment, ellipticity, skew, etc.).

In view of the uncertainty of the conditions characterizing the thermal resistance of the contact zone, calculation of heat removal through the shaft can be carried out only approximately.

For a more complete idea of the role of the shaft in removing armature losses, calculations were performed for the limiting cases: \(\delta_v = 0\) and \(\delta_v =\) [[unclear: value not visible]]. The temperature values obtained in this case showed discrepancies not exceeding 2–2.5%, while the character of the temperature field remained practically unchanged (Fig. 2).

According to comparative calculations, heat removal through the slot wedge affects mainly only the temperature level of the winding (Fig. 2). The error in calculating \(\vartheta_{a\,\mathrm{av}}\) in this case reaches 5–6%. Neglecting in the calculations the dependence of the copper losses on the nonuniformity of temperature along the length of the winding introduces an error of approximately 0.1–0.2%.

The average temperature rise of the armature winding can be obtained from the following expression:

\[ \vartheta_{a\,\mathrm{av}} = \frac{ \displaystyle \int_{-0.5l}^{-(0.5l_p+l_i)} \vartheta_{tk\,\mathrm{av}}\,dx + \int_{-(0.5l_p+l_i)}^{-0.5l_p} \vartheta_{p\,\mathrm{av}}\,dx + \int_{-0.5l_p}^{0.5l_p} \vartheta_p\,dx + \int_{0.5l_p}^{0.5l_p+l_i} \vartheta_l\,dx + \int_{0.5l_p+l_i}^{0.5l} \vartheta_t\,dx }{l} \]

Here, taking \(l_c = l_t + l_i\),

\[ \vartheta_{tk\,\mathrm{av}} = \frac{f_{tk}}{f_p}\vartheta_{tk} + \frac{f_c}{f_p}\vartheta_n + \frac{f_c}{f_p}\vartheta_v, \qquad \vartheta_{l\,\mathrm{av}} = \frac{f_l}{f_p}\vartheta_l + \frac{f_c}{f_p}\vartheta_n + \frac{f_c}{f_p}\vartheta_v . \]

The average temperature rise of the armature stack is determined from the relation:

\[ \vartheta_{\mathrm{Fe}\,\mathrm{av}} = K_c \left( q_c\frac{\Delta_p}{\Pi \lambda_p} + \frac{1}{l_p} \int_{-0.5l_p}^{0.5l_p}\vartheta_p\,dx \right). \]

Conclusion

The analytical relations presented above reflect, with sufficient accuracy in both quantitative and qualitative terms, the physical phenomenon under consideration: the temperature distribution along the armature winding of enclosed non-ventilated DC machines under steady-state thermal conditions. In view of this, the developed calculation method can be recommended for use in analytical studies of the influence of various factors on the mean level and the character of the temperature distribution along the armature winding.

References

1. Yu. P. Shlykov, E. A. Ganin. Contact Heat Transfer, Gosenergoizdat, 1963.

2. A. I. Borisenko, A. I. Yakovlev. A method for the thermal calculation of high-speed DC electric motors, Izvestiya vuzov, “Elektromekhanika,” 1965, No. 5.

3. S. S. Kutateladze, V. M. Borishanskii. Handbook of Heat Transfer, Gosenergoizdat, 1959.

4. M. N. Ulyanitskii. Investigation of Heating in Enclosed DC Machines with Natural Cooling, Dissertation, TPI, 1966.

5. M. N. Ulyanitskii. On the thermal calculation of enclosed electrical machines with natural cooling, Elektrichestvo, 1965, No. 5.