ON THE CALCULATION OF THE TEMPERATURE FIELD IN ELECTRICAL WINDINGS

G. F. Shilin, V. V. Ivanov

(Presented by Prof. Dr. G. I. Fuks)

In designing a number of electrical devices—magnetizing windings of transformers and betatrons, excitation coils of generators, etc.—it is necessary to know the temperature regime of such current-carrying elements, as well as the time required for individual sections to warm up to a prescribed temperature. New designs of electrical windings, extending the operating characteristics of existing equipment, as well as safe overload limits, are often determined by a fixed value of the permissible temperature for the winding material.

We shall seek the temperature field in an electrical winding of continuous rectangular profile, placed in a medium with zero temperature. The equivalent coefficients of thermal conductivity \(\lambda\), thermal diffusivity \(a\), and heat-transfer coefficients \(\alpha_x\) and \(\alpha_y\) are constant quantities. The values of the equivalent coefficients \(\lambda\) and \(a\) for electrical windings can be calculated by the method set forth in [1]. If it is assumed that the electrical resistance of the winding depends linearly on temperature,

\[ R = R_0(1+\gamma \cdot t), \]

where \(R_0\) is the resistance at \(0^\circ\text{C}\), \(\gamma\) is a constant, then the Joule heat, proportional to the square of the current and to the resistance, will also be a linear function of temperature,

\[ q_v = q_{v0}\cdot(1+\gamma\cdot t)\ \frac{\text{W}}{\text{m}^3}. \]

Then the system of equations describing the nonstationary heat-conduction process is written as follows:

\[ \frac{1}{a}\frac{\partial t(x,y,z)}{\partial z} = \frac{\partial^2 t(x,y,z)}{\partial x^2} + \frac{\partial^2 t(x,y,z)}{\partial y^2} + \gamma^2\cdot t(x,y,z) + B, \tag{1} \]

\[ \frac{\partial t(R_x,y,z)}{\partial x} + \frac{\alpha_x}{\lambda}\cdot t(R_x,y,z) = 0, \tag{2} \]

\[ \frac{\partial t(x,R_y,z)}{\partial y} + \frac{\alpha_y}{\lambda}\cdot t(x,R_y,z) = 0, \tag{3} \]

\[ \frac{\partial t(0,y,z)}{\partial x}=0, \qquad \frac{\partial t(x,0,z)}{\partial y}=0, \tag{4} \]

\[ t(x,y,0)=0, \tag{5} \]

\[ -R_x \le x \le R_x,\qquad -R_y \le y \le R_y,\qquad 0 \le z < \infty,\qquad \gamma^2=\frac{q_{v0}}{\lambda}\gamma,\qquad B=\frac{q_{v0}}{\lambda}. \]

We construct the desired solution in the form of a series

\[ t(x,y,\tau)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} (A_n\cos\mu_n x+B_n\sin\mu_n x)(A_m\cos p_m y+B_m\sin p_m y)\times \]

\[ \times \left[1-\exp(-\kappa_{n,m}^{2}a\tau)\right], \]

in which \(A_n; A_m; B_n; B_m; \mu_n; p_m; \kappa_{n,m}\) are constants to be determined. Taking into account that the function \(t(x,y,\tau)\) is even with respect to \(x\) and \(y\) (condition (4)), and using (2) and (3), we find

\[ t(x,y,\tau)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} A_n\cdot A_m\cos\mu_n x\cdot\cos p_m y \left[1-\exp(-\kappa_{n,m}^{2}\cdot a\cdot \tau)\right]. \tag{6} \]

Here \(\mu_n\) and \(p_m\) are, respectively, the roots of the transcendental equations

\[ \mu_n\tan\mu_n=\frac{\alpha_x}{\lambda}, \qquad p_m\tan p_m=\frac{\alpha_y}{\lambda}, \]

given in [2].

Substitution of (6) into (1) gives

\[ A_{n,m}=A_n\cdot A_m= \frac{ B\cdot\displaystyle\int_{0}^{R_x}\int_{0}^{R_y} \cos\mu_n x\cdot\cos p_m y\cdot dx\cdot dy }{ \kappa_{n,m}^{2}\displaystyle\int_{0}^{R_x}\int_{0}^{R_y} \cos^{2}\mu_n x\cdot\cos^{2}p_m y\cdot dx\cdot dy }, \tag{7} \]

where

\[ \kappa_{n,m}^{2}=\mu_n^{2}+p_m^{2}-\beta^{2}. \]

Table 1

Calculating (7), we finally obtain

\[ t(x,y,\tau)= \]

\[ =4\sum_{n=1}^{\infty}\sum_{m=1}^{\infty} \frac{ \left[\mu_n^{2}+\left(\frac{\alpha_x}{\lambda}\right)^{2}\right] \left[p_m^{2}+\left(\frac{\alpha_y}{\lambda}\right)^{2}\right] \sin\mu_n R_x\cdot\sin p_m R_y }{ \left\{R_x\left[\mu_n^{2}+\left(\frac{\alpha_x}{\lambda}\right)^{2}\right] +\frac{\alpha_x}{\lambda}\right\} \left\{R_y\left[p_m^{2}+\left(\frac{\alpha_y}{\lambda}\right)^{2}\right] +\frac{\alpha_y}{\lambda}\right\}\cdot\mu_n\cdot p_m } \times \]

\[ \times \frac{B\cdot\cos\mu_n\cdot x\cdot\cos p_m y} {\left(\mu_n^{2}+p_m^{2}-\beta^{2}\right)} \left\{1-\exp\left[-(\mu_n^{2}+p_m^{2}-\beta^{2})a\tau\right]\right\}. \tag{8} \]

For \(\alpha_x=\alpha_y=\infty,\ \tau=\infty\) (this corresponds to the stationary temperature distribution in the winding, when zero temperature is maintained on its surface), as a special case of equation (8) we obtain the solution given in [3].

Table 1 gives the calculated data for the temperature field at the center of the winding \(t(0,0,\tau)\) and on the surfaces \(t(0,R_y,\tau)\) and \(t(R_x,0,\tau)\) for various moments of time. The winding characteristics are \(R_x \times R_y = 0.04 \times 0.05\ \mathrm{m}^2\), \(\lambda = 5.815\ \mathrm{W/(m\cdot deg)}\), \(a = 13.9 \cdot 10^6\ \mathrm{m^2/sec}\), \(q_y = 139560(1+0.004\,t)\ \mathrm{W/m^3}\), \(\alpha_x = 11.63\ \mathrm{W/(m^2\cdot deg)}\), \(\alpha_y = 34.89\ \mathrm{W/(m^2\cdot deg)}\).

REFERENCES

1. G. Gotter. Heating and Cooling of Electrical Machines. Gosenergoizdat, 1961.
2. A. V. Lykov. Theory of Heat Conduction. Gostekhizdat, 1952.
3. P. Schneider. Engineering Problems of Heat Conduction. GIITL, 1960.