Analytical Study of the Influence of Various Parameters on the Level and Distribution of Temperature Along the Length of the Armature Winding 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)

According to comparative calculations [1], for \(v_k=(0.8\text{--}1.2)\) and \(2f_c/f_p<0.25\text{--}0.3\), heat exchange with the commutator does not play a significant role in the temperature distribution in the armature.\(^1\) Heat removal along the shaft and through the slot wedge also has an insignificant influence on the character of the temperature field along the winding length. On the whole, the exclusion from the thermal calculation of all the factors considered, for the indicated ratio \(\vartheta_k,\ \vartheta_{y,\mathrm{avg}},\ f_c,\ f_p\), causes a change mainly only in the average temperature level of the armature winding. Taking this into account, a number of simplifications may be introduced into the thermal calculation proposed in [1], and the influence of individual parameters on armature heating may be studied by a simplified method.

Assuming \(l_c=l_t+l_i\) and combining the winding leads to the commutator with the end connections, the armature winding may be regarded as a rod of constant cross section having five sections: the slot part, the end portions, and the portions of the end connections under the bandage insulation on the drive side and on the commutator side. In this case, the general form of the differential heat-conduction equations for the individual sections of the winding and the boundary conditions remain the same as in [1]. However, the values of certain coefficients \(A'\), \(m\) change. In accordance with the assumptions adopted, we have:

1. For the slot part of the winding,

\[ A'_1=\frac{k_c q_c+q_m}{\lambda_m f_p}, \qquad k_c=\frac{\lambda_p \Pi}{\lambda_p \Pi+\alpha b_k \Delta_p}, \qquad m_1^2=\frac{\Lambda}{\lambda_m f_p}, \qquad \Lambda=k_c\alpha b_k . \]

1. For the portions of the end connections under the bandage insulation on the drive and commutator sides,

\[ m_2^2=\frac{\Lambda_l}{\lambda_m f_p}, \qquad \Lambda_l=\frac{\alpha \lambda_l b_l}{\lambda_l+\alpha\Delta_l}, \qquad A'_2=\frac{q_m}{\lambda_m f_p}. \]

1. For the end portion of the end connections on the drive side,

\[ m_3^2=\frac{\alpha b_t}{\lambda_m f_p}, \qquad A'_3=A'_2 . \]

1. For the end portion of the end connections on the commutator side,

\[ B=A'_2 . \]

\(^1\) The conventional designations are the same as in [1].

The derivation of the formulas for calculating \(\vartheta_1,\vartheta_2,\vartheta_3,\vartheta_4\) is carried out in the same way as in [1]. As a result we have

\[ \vartheta_1 = \frac{(P_1P_5-1)(P_3+P_4)+(P_1+P_2P_3)a_1} {(P_2P_5-1)P_5a_3-P_2a_1^2}, \qquad \vartheta_2 = \frac{\vartheta_1P_5a_3-P_3-P_4}{a_1}, \]

\[ \vartheta_3=\vartheta_1\frac{1}{\operatorname{ch} m_2 l_i}+P_4, \qquad \vartheta_4=\frac{\vartheta_2+P_1}{P_4}. \]

Here

\[ a_1=\frac{m_1\operatorname{sh} m_2 l_i}{m_2\operatorname{sh} m_1 l_p}, \qquad a_2=\frac{m_3\operatorname{sh} m_2 l_i}{m_2\operatorname{ch} m_3 l_t}, \qquad a_3=\frac{\operatorname{ch} m_2 l_i-1}{\operatorname{ch} m_2 l_i}, \]

\[ P_1=a_3A_3\operatorname{sh}m_3l_t+A_2(\operatorname{ch}m_1l_i-1), \qquad P_2=a_2\operatorname{sh}m_3l_t+\operatorname{ch}m_2l_i, \]

\[ P_3=A_3(\operatorname{ch}m_2l_i-1)+a_1A_1(\operatorname{ch}m_1l_p-1), \qquad P_4=A_2a_3+\frac{Bl_t}{m_2}\operatorname{th}m_2l_i, \]

\[ P_5=a_1\operatorname{ch}m_1l_p+\operatorname{ch}m_2l_i. \]

The mean temperature rise of the armature winding is determined from the following expression:

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

To determine the level and nature of the temperature distribution along the length of the armature winding, a series of calculation variants was carried out on the Minsk-1 electronic digital computer by a simplified calculation method. The influence of the ratio of losses in copper and steel, the thermal resistance of the insulation, the design parameters of the armature, and the value of the heat-transfer coefficient was investigated. As the basic variant, the calculation was taken according to the data of the armature of a production motor of type PB42.

According to the calculated data, when the values of the electromagnetic loads are varied within the limits recommended for enclosed machines, the losses in copper have the most significant influence on the level and nature of the temperature distribution in the winding. Thus, when the current density is reduced from \(5\ \mathrm{A/mm^2}\) to \(3\ \mathrm{A/mm^2}\), \(\vartheta_{a\,\mathrm{avg}}\) decreases by approximately a factor of two, and the point with the maximum temperature shifts from the end of the end connections on the commutator side to the slotted part of the winding (Fig. 1, a). At the same time, when the induction in the teeth is varied in the range \(1.35\)--\(1.15\ \mathrm{T}\), the values of \(\vartheta_{a\,\mathrm{avg}}\) differ by no more than \(5\)--\(6\%\). In this case the position of the point with the maximum temperature does not change (Fig. 1, a).

However, on the whole, the losses in steel have a substantial influence on \(\vartheta_{a\,\mathrm{avg}}\). In machines of the type considered, the values of the coefficient \(k_c\), characterizing the influence of \(p_c\) on winding heating, according to the calculations lie within the range \(0.9\)--\(0.98\) (the upper limit applies to low-voltage machines). Thus, the losses in steel are almost completely spent on heating the copper.

The slot insulation has a comparatively weak influence on the heating of the armature winding: changing its thermal conductivity by a factor of two (from \(1.73\cdot10^{-3}\) to \(0.865\cdot10^{-3}\ \mathrm{W/(cm\,^{\circ}C)}\)) causes an increase in \(\vartheta_{a\,\mathrm{avg}}\) of only \(0.7\%\).

A somewhat greater role is played by the thermal resistance of the insulation under the bandage of the end connections. When the thermal conductivity of this insulation is reduced by a factor of two, \(\rho_{cq}\vartheta\) increases by \(4.5\%\). Change in thermal con-

Fig. 1. Calculated curves of the temperature distribution along the length of the armature winding.

1 — basic variant \((j = 5\ \mathrm{A/mm^2},\ B_z = 1.35\ \mathrm{T},\ \lambda_{\mathrm{po}}=\lambda_{\mathrm{lo}}=1.725 \cdot 10^{-3}\ \mathrm{W}/^\circ\mathrm{C}\ \mathrm{cm}^2,\ \Delta_{\mathrm{po}}=6 \cdot 10^{-2}\ \mathrm{cm},\ \Delta_{\mathrm{lo}}=5 \cdot 10^{-2}\ \mathrm{cm},\ b_{\mathrm{lo}}=1.5\ \mathrm{cm})\);

a) 2 — \(j=3\ \mathrm{A/mm^2}\), 3 — \(B_z=1.15\ \mathrm{T}\), 4 — \(\Delta_{\mathrm{p}}=2\Delta_{\mathrm{po}}\), 5 — \(\Delta_{\mathrm{l}}=2\Delta_{\mathrm{lo}}\), 6 — \(\Delta_{\mathrm{p}}=0\), 7 — \(b_{\mathrm{l}}=1.15\,b_{\mathrm{lo}}\), 8 — \(\Delta_{\mathrm{p}}=\Delta_{\mathrm{l}}=0\);

b) 2 — \(l_{\mathrm{t}}/l_{\mathrm{i}}=0.3\); 3 — \(l_{\mathrm{t}}/l_{\mathrm{i}}=0.5\);

c) 2 — \(l_{\mathrm{p}}/2l_{\mathrm{l}}=1.2\), 3 — \(l_{\mathrm{p}}/2l_{\mathrm{l}}=0.8\).

...the conductivity of the insulation within the limits considered does not lead to significant changes in the temperature distribution in the armature winding.

The heating of the armature winding depends substantially on the cooling conditions of the end connections. Thus, when their heat-transfer surface is increased by 15%, \(\vartheta_{\mathrm{a\,avg}}\) decreases by approximately 6.5%. At the same time, the ratio of the maximum and average temperatures (the nonuniformity coefficient of heating—NC) practically does not change.

With a constant length of the end connections, a decrease in the ratio \(l_{\mathrm{t}}/l_{\mathrm{i}}\) leads to a reduction in the temperature level of the winding and causes the same change in the character of the temperature field as does a decrease in the current density (Fig. 1, b). Deviation of the value of the ratio \(l_{\mathrm{t}}/l_{\mathrm{i}}\) from the average in the range of its most frequently encountered values, 0.3–0.5, leads to a change in \(\vartheta_{\mathrm{a\,max}}\) of approximately 1.6%, in \(\vartheta_{\mathrm{a\,avg}}\) of 1.3%, and in NC of 0.5%.

As the ratio \(l_{\mathrm{p}}/2l_{\mathrm{l}}\) increases, the temperature of the armature winding rises (Fig. 1, c). Moreover, deviation of the value of this ratio from the average in the range of numbers 1.4–0.6 gives a change in \(\vartheta_{\mathrm{a\,max}}\) and \(\vartheta_{\mathrm{a\,avg}}\) of approximately 2–2.5%. The value of NC in these cases remains practically constant.

A proportional increase or decrease in the length of the slot and end portions of the turn practically does not cause a change in \(\vartheta_{\mathrm{a\,avg}}\). However, \(\vartheta_{\mathrm{a\,max}}\) and NC in this case do not remain constant. With a decrease in

the coil length their values decrease; in this case the degree of change is determined by all the factors considered above.

As the heat-transfer coefficient of the armature surface increases, the nonuniformity of the temperature field increases somewhat. For example, when \(\alpha\) is increased by a factor of one and a half, the coefficient \(KH\) increases by \(2\%\).

On the whole, the results of the analytical studies carried out make it possible to draw the following conclusions:

1. The magnitude of the current density has the most substantial influence on the nature of the temperature distribution along the length of the armature winding.

2. Losses in the steel affect the heating of the armature winding to almost the same extent as losses in the copper.

3. A change in the ratios \(l_{\mathrm{T}}/l_{\mathrm{u}}\) and \(l_{\mathrm{p}}/2l_{\mathrm{p}}\) within the range of their most frequently encountered values, as well as a proportional increase (decrease) of the lengths \(l_{\mathrm{T}}, l_{\mathrm{u}}, l_{\mathrm{p}}\) by \(15\text{--}20\%\), does not exert a substantial influence on the temperature distribution along the length of the armature winding and on \(\vartheta_{\mathrm{ya\,sr}}\).

4. The position of the most highly heated point of the armature winding is not constant. Depending on the ratio of copper and steel losses and on the design parameters of the armature, the point with the maximum temperature will be located either at the end of the end connections on the commutator side, or in the slotted part of the winding between the middle of the package and its end on the commutator side.

5. The thermal resistance of the insulation of the armature winding is approximately an order of magnitude less than the resistance to heat transfer from the armature surface.

References

1. M. N. Ulyanitsky, V. V. Salomatov. On the thermal calculation of the armature of enclosed non-ventilated direct-current machines. Article in the present collection.