Approximate Calculation of the Temperature Field Along the Armature Winding Length of an Enclosed DC Machine

M. N. Ulyanitskii, V. V. Salamatov

(Recommended by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering.)

The armature of a DC machine is a system of bodies with mutual thermal coupling and different cooling conditions. Because of the complex boundary conditions, a rigorous analytical calculation for determining the temperature in the armature winding is impossible. Therefore, to simplify the problem, the following assumptions are adopted:

1. The thermal conductivity of the pack in the axial direction is neglected [1].
2. The temperature over the height of the slot and tooth is assumed constant [2].
3. The temperature distribution along the length of the winding is symmetric with respect to the middle of the pack [2].
4. Heat removal through the shaft is not taken into account.
5. The average temperature of the frame is taken as the ambient temperature.

In accordance with the above assumptions, we obtain the calculation scheme shown in Fig. 1.

Fig. 1. Calculation scheme of the armature winding.

The calculation of the temperature field in the end portion of the winding reduces to solving the differential equation

\[ \frac{d^2 \vartheta_{\mathrm{л}}}{dx^2} - n^2 \vartheta_{\mathrm{л}} + \frac{q_m}{\lambda_m f_m} = 0 \tag{1} \]

with allowance for the boundary conditions:

\[ \text{for } x=\frac{l}{2}; \quad \frac{d\vartheta_{\mathrm{л}}}{dx}=0, \qquad \text{for } x=\frac{l_a}{2}; \quad \vartheta_{\mathrm{л}}=\vartheta_0. \tag{2} \]

Integration of the system (1), (2) gives the expression for the temperature distribution along the length of the end portion:

\[ \vartheta_{\mathrm{л}} = C_1 e^{nx} + C_2 e^{-nx} + K, \tag{3} \]

where

\[ C_1= \frac{\vartheta_0-K}{2\,\operatorname{ch} n l_{\mathrm{л}}} e^{-n\frac{l}{2}}; \qquad C_2= \frac{\vartheta_0-K}{2\,\operatorname{ch} n l_{\mathrm{л}}} e^{n\frac{l}{2}}; \]

\[ n^2= \frac{\Lambda_{\mathrm{л}}}{\lambda_m f_m}; \qquad \Lambda_{\mathrm{л}}= \frac{\alpha u \lambda_{\mathrm{л}}}{\lambda_{\mathrm{л}}+\alpha \Delta}; \qquad K= \frac{q_m}{\Lambda_{\mathrm{л}}}. \]

Solving the differential equation for the temperature distribution along the length of the slot part of the winding

\[ \frac{d^{2}\vartheta_{\text{п}}}{dx^{2}}-m^{2}\vartheta_{\text{п}}+A=0 \tag{4} \]

with allowance for the boundary conditions

\[ \text{at}\qquad x=0;\qquad \frac{d\vartheta_{\text{п}}}{dx}=0, \qquad x=\frac{l_a}{2};\qquad \vartheta_{\text{п}}=\vartheta_0 \tag{5} \]

gives the following result:

\[ \vartheta_{\text{п}}= \left(\vartheta_0-\frac{A}{m^2}\right) \frac{\operatorname{ch} mx}{\operatorname{ch} m\frac{l_a}{2}} +\frac{A}{m^2}, \tag{6} \]

where

\[ A=\frac{\kappa_c q_c+q_m}{\lambda_m f_m}; \qquad \kappa_c= \frac{\lambda_{\text{п}}\dfrac{\Pi}{\Delta_{\text{п}}}} {ab+\lambda_{\text{п}}\dfrac{\Pi}{\Delta_{\text{п}}}}; \]

\[ m^2=\frac{\Lambda_{\text{п}}}{\lambda_m f_m}; \qquad \Lambda_{\text{п}}=\kappa_c ab. \]

To determine \(\vartheta_0\), we use the condition of thermal contact in the winding section \(x=\dfrac{l_a}{2}\)

\[ \frac{d\vartheta_{\text{п}}}{dx}= \frac{d\vartheta_{\text{л}}}{dx}. \tag{7} \]

From expressions (3), (6), (7) we have

\[ \vartheta_0= \frac{\dfrac{A}{m}\operatorname{th} m\frac{l_a}{2} +nK\operatorname{th} nl_{\text{л}}} {m\operatorname{th} m\frac{l_a}{2} +n\operatorname{th} nl_{\text{л}}}. \tag{8} \]

Experimental verification of the relations presented has shown sufficiently close agreement between experimental and calculated data. This provides grounds for recommending the use of dependences (3), (6), (8) in engineering calculations for an approximate assessment of the thermal stress of the armature insulation of enclosed direct-current machines.

Symbols

\(q_m, q_c\) — specific losses in the winding copper and in the armature steel

\[ q_m=\frac{P_m}{zl}; \qquad q_c=\frac{P_c}{zl_a}. \]

\(P_m, P_c\) — losses in the winding copper and in the armature steel;
\(z\) — number of armature slots;
\(a\) — heat-transfer coefficient of the armature surface;
\(f_m\) — cross-sectional area of the copper in the slot;
\(\Pi\) — slot perimeter;
\(u\) — width of the end part of the section

\[ u=\frac{\pi D_{\ell}}{z}; \]

\(D_{\ell}\) — outer diameter of the end connections of the armature winding;
\(\Delta_{\mathrm{p}}\) — thickness of the slot insulation;
\(\Delta_{\ell}\) — thickness of the insulation under the bandage;
\(b\) — tooth width along the outer diameter of the stack, with twice the slot height;
\(\lambda_m, \lambda_{\mathrm{p}}, \lambda_{\ell}\) — thermal conductivity of copper, of the slot insulation, and of the insulation under the bandage;
\(\ell_{\ell}\) — half the length of the end part.

REFERENCES

1. I. M. Postnikov. Design of Electrical Machines. GITL, Ukrainian SSR, 1960.

2. M. N. Ulyanitskii. Analysis of temperature fields of series-P direct-current machines. Proceedings of TPI, vol. 132, 1965.