METHOD FOR MEASURING THE THERMAL CONDUCTIVITY OF STATOR WINDING INSULATION

M. A. SANNIKOVA, D. I. SANNIKOV

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

The method is based on measuring the temperature drop in the winding insulation at a known heat flux. The winding is heated by direct current. At the same time, the power of heat release and the average temperature of the winding are determined from the resistance. The temperature of the core and of the surface of the end portions is measured by a contact method using thermocouples pressed tightly against the outer and inner surfaces of the stator core and the end portions. The average temperature of the steel is determined by averaging the thermocouple readings. In this case, the nonuniformity of steel heating around the stator circumference causes an error in determining the average temperature; however, this error decreases as the number of thermocouples increases. Knowing from experience the scatter of steel temperatures and setting a certain permissible error, one can, by methods of mathematical statistics, calculate the necessary number of thermocouples installed both in the bore and on the back of the stator. In our case the number of thermocouples is 64.

The calculation of the thermal conductivity of the insulation from the measurement data is carried out on the basis of the equivalent thermal circuit of the stator, shown in Fig. 1.

Fig. 1.

Here

\(P\) is the losses in the winding due to the heating current;

\[ \varepsilon=\frac{l_{\mathrm{p}}}{l_{\mathrm{p}}+l_{\mathrm{e}}} \]

is the relative length of the slot part;

\(\varepsilon P\) and \((1-\varepsilon)P\) are the losses in the active and end parts;

\(\Theta_{\mathrm{e}}\) and \(\Theta_{\mathrm{a}}\) are the mean temperatures of the end and active parts;

\(\Theta_1\) is the mean temperature of the steel around the slots, measured by thermocouples;

\(\Theta_1+\Delta\Theta_1\) is the temperature of this same part with some error \(\Delta\Theta_1\);

\(\alpha\) is the fraction of this uncontrolled surface in the total surface of the steel around the slots;

\(\Lambda_{\mathrm{a}}\) is the thermal conductance of the insulation in the slot part (taking into account turn and ground insulation);

\(\Lambda_{\mathrm{a}}(1-\alpha)\) is the thermal conductance of the controlled part of the slot surface;

\(\Lambda_{\mathrm{a}}\alpha\) is the thermal conductance of the uncontrolled part.

Similarly, for the end parts their air-cooling surface can be divided into controlled and uncontrolled:

\(\Theta_2\) is the temperature of the controlled surface;

\(\Delta\Theta_2\) is the temperature error on the uncontrolled surface;

\(\lambda\) is the fraction of the uncontrolled surface;

\(\Lambda_{\mathrm{e}}\) is the thermal conductance of the bundle of conductors in the end part of the winding from the middle to the surface;

\(\Lambda_{\mathrm{e}}(1-\lambda)\) is the conductance to the controlled part of the surface;

\(\Lambda_{\mathrm{e}}\lambda\) is the conductance to the uncontrolled part;

\(\Lambda_{\mathrm{m}}\) is the thermal conductance along the conductors between the end parts and the slot part. It is determined by calculation on the basis of grid modeling.

For the equivalent circuit we write the heat-balance equation

\[ \left\{ \begin{aligned} \theta_{\mathrm{a}}\left[\Lambda_{\mathrm{m}}+\Lambda_{\mathrm{a}}(1-\alpha)+\Lambda_{\mathrm{a}}\alpha\right] &=\varepsilon P+\theta_1\Lambda_{\mathrm{a}}(1-\alpha)+(\theta_1+\Delta\theta_1)\Lambda_{\mathrm{a}}\alpha,\\ -\theta_{\mathrm{a}}\Lambda_{\mathrm{m}}+\theta_{\mathrm{e}}\left[\Lambda_{\mathrm{m}}+\Lambda_{\mathrm{e}}(1-\lambda)+\Lambda_{\mathrm{e}}\lambda\right] &=(1-\varepsilon)P+\theta_2\Lambda_{\mathrm{e}}(1-\lambda)+(\theta_2+\Delta\theta_2)\Lambda_{\mathrm{e}}\lambda . \end{aligned} \right. \tag{1} \]

Solving the system with respect to \(\Theta_{\mathrm{a}}\) and \(\Theta_{\mathrm{e}}\), we obtain:

\[ \theta_{\mathrm{a}}=\frac{1}{D}\left\{ P[\varepsilon\Lambda_{\mathrm{e}}+\Lambda_{\mathrm{m}}] +\theta_1\Lambda_{\mathrm{a}}(\Lambda_{\mathrm{e}}+\Lambda_{\mathrm{m}}) +\theta_2\Lambda_{\mathrm{e}}\Lambda_{\mathrm{m}} +\Delta\theta_1\alpha\Lambda_{\mathrm{a}}(\Lambda_{\mathrm{e}}+\Lambda_{\mathrm{m}}) +\Delta\theta_2\lambda\Lambda_{\mathrm{e}}\Lambda_{\mathrm{m}} \right\}, \tag{2} \]

\[ \theta_{\mathrm{e}}=\frac{1}{D}\left\{ P[(1-\varepsilon)\Lambda_{\mathrm{a}}+\Lambda_{\mathrm{m}}] +\theta_1\Lambda_{\mathrm{a}}\Lambda_{\mathrm{m}} +\theta_2\Lambda_{\mathrm{e}}(\Lambda_{\mathrm{a}}+\Lambda_{\mathrm{m}}) +\Delta\theta_1\alpha\Lambda_{\mathrm{a}}\Lambda_{\mathrm{m}} +\Delta\theta_2\lambda\Lambda_{\mathrm{e}}(\Lambda_{\mathrm{a}}+\Lambda_{\mathrm{m}}) \right\}. \]

Here \(D=\Lambda_{\mathrm{a}}\Lambda_{\mathrm{e}}+\Lambda_{\mathrm{a}}\Lambda_{\mathrm{m}}+\Lambda_{\mathrm{e}}\Lambda_{\mathrm{m}}\) is the determinant of the system of original equations.

We introduce the notation:

\[ R_p=\frac{1}{D}\left[\varepsilon^2\Lambda_{\mathrm{e}}+(1-\varepsilon)^2\Lambda_{\mathrm{a}}+\Lambda_{\mathrm{m}}\right]; \tag{3} \]

\[ k_1=\frac{1}{D}\Lambda_{\mathrm{a}}(\varepsilon\Lambda_{\mathrm{e}}+\Lambda_{\mathrm{m}}); \tag{4} \]

\[ k_2=\frac{1}{D}\Lambda_{\mathrm{e}}\left[(1-\varepsilon)\Lambda_{\mathrm{a}}+\Lambda_{\mathrm{m}}\right]=1-k_1. \tag{5} \]

The mean temperature of the winding

\[ \theta_{\mathrm{av}}=\varepsilon \theta_a+(1-\varepsilon)\theta_l . \tag{6} \]

Taking (2), (3), (4), and (5) into account,

\[ \theta_{\mathrm{av}}=P R_p+\theta_1 k_1+\theta_2(1-k_1)+\Delta\theta_1 \alpha k_1+\Delta\theta_2 \lambda(1-k_1). \tag{7} \]

The temperature difference is determined as

\[ \Delta\theta_{\mathrm{av}}=\theta_{\mathrm{av}}-\theta_1 = P R_p+(\theta_2-\theta_1)(1-k_1)+\partial\theta, \tag{8} \]

where

\[ \partial\theta=\Delta\theta_1 \alpha k_1+\Delta\theta_2 \lambda(1-k_1) \]

is the error in determining the difference.

On the basis of several measurements under different cooling conditions, we determine \(R_p\) and \(k_1\) from the graph (Fig. 2). The graph is constructed with a gradual change in the stator ventilation rate and \(P=\mathrm{Const}\). Initially we neglect the error \(\partial\theta\).

From the graph, at \(\Theta_2-\Theta_1=0\),

\[ \Delta\theta_{\mathrm{av}\,10}=P R_p \quad \text{and} \quad R_p=\frac{\Delta\theta_{\mathrm{av}\,10}}{P}. \]

Fig. 2.

We choose an arbitrary point \(b\); then

\[ \Delta\theta'=(\theta_2-\theta_1)_b(1-k_1)=(\theta_2-\theta_1)_b k_2, \]

whence

\[ k_2=\frac{\Delta\theta'}{(\theta_2-\theta_1)_b},\qquad k_1=1-k_2. \]

To reduce the influence of the error \(\partial\theta\), the experiment should be repeated several times under different conditions, for example, with additional thermal insulation of the end parts and of the stack (not simultaneously): \(R_p\) and \(k_2\) are taken as the averages from these experiments. If \(R_p\) and \(k_1\) have been found, then the calculation of \(\Lambda_a\) and \(\Lambda_l\) presents no difficulty.

For simplicity, we introduce thermal resistances instead of conductivities:

\[ R_a=\frac{1}{\Lambda_a};\qquad R_l=\frac{1}{\Lambda_l};\qquad R_m=\frac{1}{\Lambda_m}. \]

Taking into account equations (3), (4), (5), (7) and the replacement of conductivities by thermal resistances, we obtain the system of equations (9):

\[ \begin{cases} k_1 R_a+(1-k_1)R_l=(\varepsilon-k_1)R_m,\\ \varepsilon k_1 R_a+(1-\varepsilon)(1-k_1)R_l=R_p. \end{cases} \tag{9} \]

Solving equation (9) with respect to \(R_a\) and \(R_l\), we find:

\[ R_a = -\frac{1}{k_1}\left[R_p + (1-\varepsilon)(\varepsilon-k_1)R_M\right], \]

\[ R_l = \frac{1}{1-k_1}\left[R_p - \varepsilon(\varepsilon-k_1)R_M\right] \tag{10} \]

or

\[ \Lambda_a = \frac{k_1}{R_p + (1-\varepsilon)(\varepsilon-k_1)R_M}, \tag{11} \]

\[ \Lambda_l = \frac{1-k_1}{R_p - \varepsilon(\varepsilon-k_1)R_M}. \tag{12} \]

Preliminary calculations show that the possible error in determining the conductivity lies within the range of 5–10%.