TWO-DIMENSIONAL TEMPERATURE FIELD IN AN OVAL OR TRAPEZOIDAL SLOT OF AN ELECTRICAL MACHINE

V. A. ZHADAN, D. I. SANNIKOV

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

Calculation of the steady-state temperature field in the slot of an electrical machine with a winding made of round wire makes it possible to estimate the nonuniformity of the thermal stress of the insulation of individual conductors, to identify the principal causes of this nonuniformity, and to refine methods for calculating the equivalent parameters of the thermal coupling of the winding with the core and the cooling air.

The calculation is based on the following assumptions:

1) the slot, excluding the slot insulation, is considered to be filled with a homogeneous mass with equivalent thermal conductivity \(\lambda_{\mathrm{e}}\) and uniformly distributed losses \(w\);

2) heat transfer along the winding conductors is not taken into account;

3) variation of losses in different conductors as a result of their different temperatures is not taken into account; the volumetric loss density is taken at the mean temperature of the winding in the cross section under consideration.

A universal method for solving such problems in calculating a two-dimensional field is the grid, or cell, method. Figure 1 gives, as an example, the subdivision into cells of one sector of the tooth zone for the armature of a P42 motor, 4.5 kW, 1500 rpm. The calculation was performed taking into account

Fig. 1. Calculation of the temperature field in a slot by the cell method.
1–4—thermocouple installation points.

thermal conductivity and heat transfer of the wedge. The temperature distribution is shown by means of isotherms. As can be seen from the figure, it is characterized by considerable nonuniformity; as the slot widens upward, the temperature correspondingly rises; however, the heat removal through the wedge and the temperature drop over the height of the tooth cause the hot spot to shift toward the middle of the slot.

Calculation by the grid method is very complicated and labor-intensive; therefore a simpler method may be proposed, based on an approximate analytical solution of the heat-conduction equation. The temperature distribution over the width of the slot is taken to be parabolic, and over the width of the tooth—constant, as a result of which the calculation of the two-dimensional field is reduced to the solution of a one-dimensional problem.

The density of the heat flux passing from the slot into the tooth through the lateral surface is

\[ A = (t_{\mathrm{p}} - t_z)\,Y . \]

Here \(t_{\mathrm{p}}\) is the temperature averaged over the width of the slot;
\(t_z\) is the temperature in the tooth section;

\[ Y = \frac{1}{\dfrac{\Delta_{\mathrm{i}}}{2\lambda_{\mathrm{i}}}+\dfrac{b}{12\lambda_{\mathrm{e}}}} \tag{1} \]

is the transverse conductance per unit height of the slot for a parabolic temperature distribution over the width, where \(\Delta_{\mathrm{i}}\) is the thickness of the insulation;
\(b\) is the width of the slot excluding the slot insulation;
\(\lambda_{\mathrm{i}}\) is the averaged thermal conductivity of the layered slot insulation.

Taking the axis of symmetry of the slot as the coordinate axis \(x\), measured from the wedge to the bottom of the slot, on the basis of the heat balance for an element \(dx\) of the tooth zone (including the slot and the tooth), one can write the system of differential heat-conduction equations

\[ \frac{d}{dt}\left(g\,\frac{dt_{\mathrm{p}}}{dx}\right) - (t_{\mathrm{p}} - t_z)Y + \sigma = 0. \tag{2} \]

\[ g_z\,\frac{d^2 t_z}{dx^2} \pm (t_{\mathrm{p}} - t_z)Y + \sigma_z = 0, \tag{3} \]

where \(g=b\lambda_{\mathrm{e}};\ \sigma=bw;\)

\(g_z\) is the conductance and \(\sigma_z\) the losses per unit height of the tooth. For a trapezoidal slot with maximum width \(b_0\) at \(x=0\) and minimum width \(b_h\) at \(x=h\),

\[ b=b_0(1-k_0x); \tag{4} \]

\[ Y=\frac{Y_0}{1-c_0x}, \tag{5} \]

where

\[ k_0=\frac{b_0-b_h}{hb_0}; \tag{6} \]

\[ c_0=Y_0\,\frac{b_0k_0}{12\lambda_{\mathrm{e}}}; \tag{7} \]

\(h\) is the height of the part of the slot occupied by the winding;
\(Y_0\) is the value of the transverse conductance at \(b=b_0\). For an oval slot, the same formulas may be applied provided it is replaced by an equivalent trapezoidal slot.

Thus, equations (2) and (3) have variable coefficients, and their solution is not expressible in tabulated functions, which makes it of little use for practical application.

The following solution procedure is the most convenient. First, a rectangular slot is considered \((b=\mathrm{const})\); on the basis of an analysis of this case, the temperature is divided into several components, each of which has a definite physical meaning; proceeding from this, the corresponding components for \(b=\mathrm{Var}\) are calculated in simplified form.

The general solution of the problem for the case \(b=\mathrm{const}\) is given in [1]. In view of the considerable difference between the longitudinal conductivities of the tooth and the slot, \(g_z \gg g\), the expressions obtained in [1] for the temperatures can be substantially simplified:

\[ t_z=t_{z0}\frac{h-x}{h}+t_{zh}\frac{x}{h}+\frac{\sigma+\sigma_z}{2g_z}x(h-x), \tag{8} \]

\[ t_n=t_z+t_\sigma+(t_{n0}-t_{z0}-t_\sigma)e^{-\beta x} +(t_{nh}-t_{zh}-t_\sigma)e^{-\beta(h-x)}. \tag{9} \]

Here \(t_{z0}, t_{zh}, t_{n0}\), and \(t_{nh}\) are boundary values of the temperature;

\[ t_\sigma=\frac{\sigma}{Y} \]

is the excess of the winding temperature over the tooth temperature in the absence of heat flow in the slot along the \(x\)-axis;

\[ (t_{n0}-t_{z0}-t_\sigma)e^{-\beta x} \quad \text{and} \quad (t_{nh}-t_{zh}-t_\sigma)e^{-\beta(h-x)} \]

are exponential components arising as a result of changes in \(t_{n0}\) and \(t_{nh}\) in the presence of heat discharge through the wedge and the insulation of the slot bottom;

\[ \beta=\sqrt{\frac{Y}{g}} \]

is the propagation constant of the temperature field in the slot along the \(x\)-axis.

In the case of a trapezoidal slot, in the expression for \(t_z\) only the component proportional to the losses in the slot changes:

\[ t_{z\sigma}=\frac{\sigma}{2g_z}x(h-z), \]

which corresponds to the temperature distribution in a rod of length \(h\) with losses \(\sigma\) distributed along its length, under the condition \(t_{z0}=t_{zh}=0\). Numerical calculations for real ranges of variation of the thermal parameters show that sufficient accuracy (within \(0.1^\circ\)) is ensured by substituting in (8)

\[ \sigma=\frac{\sigma_0+\sigma_h}{2}. \]

To determine the character of the component \(t_\sigma\) for a trapezoidal slot shape, it is necessary to consider the case in which all the other components are absent, that is, \(t_z=0\), and the boundary conditions are such as if the slot were continued in both directions beyond the interval \(0<x<h\) to such distances that the boundary conditions at its new ends practically do not affect the field within the interval \(0<x<h\). In this case the field can be calculated by the method of successive approximations.

In the first approximation, heat motion along the \(x\)-axis is not taken into account,

\[ t_\sigma'=\frac{\sigma}{Y}=\frac{\sigma_0}{Y_0}(1-k_0x)(1-c_0x). \tag{10} \]

In the second approximation, the heat inflow to each point due to the nonuniformity of the temperature distribution \(t_{\sigma}'\) is taken into account,

\[ \Delta \sigma = \frac{d}{dx}\left(g\frac{dt_{\sigma}'}{dx}\right) = \sigma_0 k_0 \frac{g_0}{Y_0} \left(k_0+3c_0-4k_0c_0x\right). \tag{11} \]

The additional temperature rise is

\[ \Delta t_{\sigma} = -\frac{\Delta \sigma}{Y} = -\frac{\sigma_0 k_0 g_0}{Y_0^2} \left(k_0+3c_0-4k_0c_0x\right)(1-c_0x). \tag{12} \]

The additional heat inflow in the third approximation amounts to fractions of a percent of \(\sigma\) and may be disregarded.

The final expression for \(t_{\sigma}\) is

\[ t_{\sigma} = t_{\sigma 0}\frac{h-x}{h} + t_{\sigma h}\frac{x}{h} - \frac{\sigma_0}{Y_0} k_0 c_0 \left(1+4k_0c_0\frac{g_0}{Y_0}\right) x(h-x)k, \tag{13} \]

where

\[ t_{\sigma 0} = \frac{\sigma_0}{Y} \left[ 1+k_0\frac{g_0}{Y_0}(k_0+3c_0) \right]; \tag{14} \]

\[ t_{\sigma h} = \frac{\sigma_h}{Y_h} \left[ 1+k_h\frac{g_h}{Y_h}(k_h+3c_h) \right]; \tag{15} \]

moreover

\[ k_h=-\frac{k_0}{1-k_0h}; \qquad c_h=-\frac{c_0}{1-c_0h}. \tag{16} \]

The influence of the boundary conditions on the temperature distribution over the slot height is manifested in the appearance of the components

\[ t_{\varepsilon 0} = (t_{n0}-t_{z0}-t_{\sigma 0})\,f_0(x) \tag{17} \]

and

\[ t_{\varepsilon h} = (t_{nh}-t_{zh}-t_{\sigma h})\,f_h(h-x). \tag{18} \]

The function \(f_0(x)\) decreases from the end \(x=0\) to the opposite end practically to zero; the function \(f_h(h-x)\) similarly decreases from the end \(x=h\). In view of the considerably slower variation of the parameters \(q\) and \(Y\) in comparison with the functions \(f_0(x)\) and \(f_h(h-x)\), the latter may be replaced by exponentials,

\[ f_0(x)=e^{-\beta_{0x}x}, \qquad f_h(h-x)=e^{-\beta_{hx}(h-x)}, \tag{19} \]

with variable coefficients:

\[ \beta_{0x} = \sqrt{\frac{Y}{g}} = \sqrt{ \frac{Y_0}{g_0} \frac{1}{(1-k_0x)(1-c_0x)} } \approx \beta_0\left(1+\frac{k_0+c_0}{2}x\right), \tag{20} \]

where

\[ \beta_0=\sqrt{\frac{Y_0}{g_0}}; \]

similarly,

\[ \beta_{hx} = \beta_h \left[ 1+\frac{k_h+c_h}{2}(h-x) \right]. \tag{21} \]

Thus, the temperature distribution over the slot height has the following expression:

\[ t_n=t_z+t_{\sigma}+t_{\varepsilon 0}+t_{\varepsilon h}. \tag{22} \]

The boundary temperatures are determined taking into account the connection of the considered section of the tooth zone of height \(h\) with the armature back, on the one hand, and with the tooth crown and wedge, on the other, on the basis of the expression for the boundary heat fluxes directed from the tooth zone into the adjacent regions.

\[ q_{n0}=-\frac{1}{g_0}\frac{dt_n}{dx}\quad (x=0); \tag{23} \]

\[ g_{nh}=-\frac{1}{g_h}\frac{dt_n}{dx}\quad (x=h); \]

for \(q_{z0}\) and \(q_{zh}\) similarly.

After the corresponding calculations and simplifications, a system of equations is obtained:

\[ \begin{vmatrix} q_{n0}\\ q_{nh}\\ q_{z0}\\ q_{zh} \end{vmatrix} = \begin{Vmatrix} -(\Lambda_0+\Lambda_0') & 0 & \Lambda_0 & \Lambda_0'\\ 0 & -(\Lambda_h+\Lambda_h') & \Lambda_h' & \Lambda_h\\ \Lambda_0 & \Lambda_h' & -(\Lambda_0+\Lambda_z+\Lambda_h') & \Lambda_z\\ \Lambda_0' & \Lambda_h & \Lambda_z & -(\Lambda_z+\Lambda_h+\Lambda_0') \end{Vmatrix} \times \begin{vmatrix} t_{n0}\\ t_{nh}\\ t_{z0}\\ t_{zh} \end{vmatrix} + \begin{vmatrix} p_{n0}\\ p_{nh}\\ p_{z0}\\ p_{zh} \end{vmatrix}; \tag{24} \]

where

\[ \Lambda_z=\frac{g_z}{h}; \]

\[ \Lambda_0=g_0(\beta_0-1/h);\qquad \Lambda_h=g_h(\beta_h-1/h); \]

\[ \Lambda_0'=\frac{g_0}{h};\qquad \Lambda_h'=\frac{g_h}{h}; \]

\[ p_{n0}=\frac{\sigma_0}{\beta_0} \left( 1-\frac{k_p+c_0}{\beta_0} +k_0\frac{k_0+3c_0}{\beta_0^2} \right); \]

\[ p_{nh}=\frac{\sigma_h}{\beta_h} \left( 1-\frac{k_h+c_h}{\beta_h} +\kappa_h\frac{k_h+3c_h}{\beta_h^2} \right); \]

\[ p_{z0}=\frac{(2\sigma_0+\sigma_h+3\sigma_z)}{6}\,h-p_{n0}; \]

\[ p_{zh}=\frac{(\sigma_0+2\sigma_h+3\sigma_z)}{6}\,h-p_{nh}. \]

A comparison of the results of calculation by the proposed method with the grid method and with experiment is given in Fig. 2. In addition to the temperature \(t_n\), the graph gives the temperature distribution along the axis of symmetry of the slot

\[ t_{nM}=t_n+(t_n-t_z)Y\frac{b}{24\lambda_e}. \tag{25} \]

As can be seen from the graph, the discrepancy between the analytical and numerical methods is insignificant. The readings of thermocouples confirm the correctness of both methods within the limits of the usual scatter of experimental data.

Thus, the proposed method of approximate analytical calculation of a two-dimensional temperature field in an oval or trapezoidal slot of an electrical machine gives sufficiently high accuracy.

Fig. 2. Temperature distribution over the height of the slot and tooth.
1 — calculation by the cell method,
2 — calculation by the proposed method,
1—4 — thermocouple readings.

REFERENCES

1. D. I. Sannikov. Calculation of the temperature field in the slot of an electrical machine, Izvestiya TPI, vol. 117, 1963.