CALCULATION OF THE TEMPERATURE FIELD IN THE SLOT OF AN ELECTRIC MACHINE

D. I. Sannikov

(Presented by the scientific seminar of the Electromechanical Faculty)

Temperature distribution over the height of the tooth and slot under steady-state thermal conditions

If the winding of an electric machine is uniformly heated along its length in the slot portion, then the temperature field in the slot is two-dimensional. An exact analytical expression for the two-dimensional field in a body of rectangular cross section under steady-state thermal conditions contains functional series with the roots of a transcendental equation as coefficients [2]. In the present case the exact solution is further complicated by the influence of the teeth.

On the other hand, both for practical purposes and for theoretical analysis of machine heating, it is desirable to have sufficiently simple formulas which nevertheless take into account the influence of all the principal factors. It is therefore advisable to use an approximate method.

The calculation is based on the following assumptions:

1. The temperature in the slot \(t_{\mathrm{p}}\) depends only on the coordinate \(x\) (Fig. 1); a temperature drop occurs only in the slot insulation. This assumption is sufficiently justified for a winding made of conductors of rectangular cross section placed in the slot in one or two rows.

A noticeable variation of temperature along the coordinate \(y\) in the case of a random winding can be taken into account by a corresponding increase in the thermal resistance of the slot insulation. In this case the quantity \(t_{\mathrm{p}}\) is taken to be the average temperature across the width of the slot.

1. The temperature of the tooth \(t_{\mathrm{z}}\) depends only on the coordinate \(x\). The basis for this assumption is the comparatively high thermal conductivity of steel and the small width of the tooth in comparison with its height.

2. The slot has a homogeneous filling with uniformly distributed losses \(p_{\mathrm{m}}\) and an equivalent thermal conductivity \(\lambda_{\mathrm{e}}\), the methods for calculating which have been considered by a number of authors [2, 4].

3. The density of heat sources in the slot, \(p_{\mathrm{m}}\), and in the tooth, \(p_{\mathrm{st}}\), is constant; moreover, the dependence of \(p_{\mathrm{m}}\) on the temperature distribution in the slot is not taken into account, and the value of the specific resistance of copper \(\rho\) is taken for the average temperature in the slot.

4. The dependence of the thermal conductivity of steel \(\lambda_{\mathrm{st}}\) and \(\lambda_{\mathrm{e}}\) on temperature is not taken into account.

1. The heat flows at the boundaries of the tooth zone \(A_{\text{zo}}, A_{\text{po}}, A_{\text{zh}}, A_{\text{ph}}\) are uniformly distributed over the width of the tooth and the slot.

2. The heights of the slot and the tooth \(h\) are identical; the slot width \(b_{\text{p}}\) and the tooth width \(b_{\text{z}}\) are constant.

The schematized tooth zone obtained on the basis of the indicated assumptions is shown in Fig. 1.

All subsequent presentations refer to one slot division and to \(1\ \mathrm{cm}\) of machine length.

We introduce the following notation:

Thermal conductance per \(1\ \mathrm{cm}\) of tooth height

\[ \Lambda_{\text{z}}=\lambda_{\text{st}} b_{\text{z}}. \]

Fig. 1. Schematized tooth zone.

Thermal conductance per \(1\ \mathrm{cm}\) of slot height

\[ \Lambda_{\text{p}}=\lambda_{\text{e}} b_{\text{p}}. \]

Thermal conductance between the tooth and the armature per \(1\ \mathrm{cm}\) of height—\(\Lambda\).

Losses per \(1\ \mathrm{cm}\) of height

in the tooth—\(q_{\text{z}}=p_{\text{st}} b_{\text{z}}\),

in the slot—\(q_{\text{p}}=p_{\text{m}} b_{\text{p}}\).

Heat flow in the tooth along the \(x\)-axis—\(A_{\text{z}}\).

Heat flow in the slot along the \(x\)-axis—\(A_{\text{p}}\).

Heat flow passing from the slot into the tooth per \(1\ \mathrm{cm}\) of height, \(a\).

Let us write the energy balance for an element of the tooth and slot of height \(dx\) (Fig. 1):

\[ A_{\text{z}}+a\,dx+q_{\text{z}}\,dx=A_{\text{z}}+dA_{\text{z}}, \tag{1} \]

\[ A_{\text{p}}-a\,dx+q_{\text{p}}\,dx=A_{\text{p}}+dA_{\text{p}}, \]

whence the derivatives with respect to \(x\) are

\[ A'_{\text{z}}=q_{\text{z}}+a, \]

\[ A'_{\text{p}}=q_{\text{p}}-a. \tag{2} \]

According to Fourier’s law,

\[ t'_z=-\frac{1}{\lambda_z}A_z, \]

\[ t'_p=-\frac{1}{\lambda_p}A_p. \tag{3} \]

In addition,

\[ u=(t_p-t_z)\Lambda . \tag{4} \]

Differentiating (3) once more and taking (2) and (4) into account, we obtain a system of differential equations determining the law of temperature distribution in the slot and in the tooth along the \(x\)-axis:

\[ t''_z=-\frac{1}{\lambda_z}\left[q_z+(t_p-t_z)\Lambda\right], \]

\[ t''_p=-\frac{1}{\lambda_p}\left[q_p+(t_z-t_p)\Lambda\right], \tag{5} \]

the general solution of which has the form

\[ t_z=\frac{1}{\lambda_z}\left[C_1 e^{kx}+C_2 e^{-kx}-\frac{\lambda_z\lambda_p}{\lambda_0}t_0\right] -\frac{q}{2\lambda_0}x^2-\frac{C_3}{\lambda_0}x+\frac{C_4}{\lambda_0}, \tag{6} \]

\[ t_p=\frac{1}{\lambda_p}\left[C_1 e^{kx}+C_2 e^{-kx}-\frac{\lambda_z\lambda_p}{\lambda_0}t_0\right] -\frac{q}{2\lambda_0}x^2-\frac{C_3}{\lambda_0}x+\frac{C_4}{\lambda_0}, \]

where

\[ k=\sqrt{\frac{\Lambda(\lambda_z+\lambda_p)}{\lambda_z\lambda_p}}, \]

\[ \lambda_0=\lambda_z+\lambda_p, \]

\[ q=q_z+q_p, \]

\[ t_0=\frac{\lambda_z q_p-\lambda_p q_z}{\Lambda(\lambda_z+\lambda_p)} \]

is a quantity having the dimension of temperature.

Heat fluxes:

\[ A_z=-\lambda_z t'_z=-k\left(C_1 e^{kx}-C_2 e^{-kx}\right) +\lambda_z\frac{qx+C_3}{\lambda_0}, \]

\[ A_p=-\lambda_p t'_p=k\left(C_1 e^{kx}-C_2 e^{-kx}\right) +\lambda_p\frac{qx+C_3}{\lambda_0}. \tag{7} \]

As boundary conditions, any four quantities out of the following eight may be specified:

\[ t_{z0},\ t_{p0},\ A_{z0},\ A_{p0}\quad \text{at } x=0 \]

and

\[ t_{zh},\ t_{ph},\ A_{zh},\ A_{ph}\quad \text{at } x=h. \]

except for the four heat flows \(A_{z0}, A_{p0}, A_{zh}, A_{ph}\), since they are connected by the energy-balance equation

\[ A_{z0}+A_{p0}+qh=A_{zh}+A_{ph}. \tag{8} \]

If, as boundary conditions, the temperature of the end points is specified,

\[ t_{z0},\ t_{p0},\ t_{zh},\ t_{ph}, \]

then the following particular solution is obtained:

\[ t_z=-\frac{\Lambda_p}{\Lambda_0}(t_a+t_b)+t_c+t_d, \]

\[ t_p=\frac{\Lambda_z}{\Lambda_0}(t_a+t_b)+t_c+t_d, \tag{9} \]

where

\[ t_a=(t_{ph}-t_{zh})\frac{Shkx}{Shkh}+(t_{p0}-t_{z0})\frac{Shk(h-x)}{Shkh}, \tag{10} \]

\[ t_b=t_0\left[1-\frac{Shkx+Shk(h-x)}{Shkh}\right], \tag{11} \]

\[ t_c=\frac{1}{h\Lambda_0}\left[(\Lambda_p t_{p0}+\Lambda_z t_{z0})(h-x)+(\Lambda_p t_{ph}+\Lambda_z t_{zh})x\right], \tag{12} \]

\[ t_d=\frac{q}{2\Lambda_0}x(h-x). \tag{13} \]

Thus, the temperature of the tooth and of the slot is divided into four components, of which \(t_a\) and \(t_c\) depend only on the temperature of the end points, whereas the magnitude of the losses \(q_p\) and \(q_z\) does not enter explicitly here; \(t_b\) and \(t_d\), on the contrary, depend only on the losses in the tooth and in the slot. The same applies also to the heat flows \(A_z\) and \(A_p\); indeed, on the basis of (7) and (9),

\[ A_z=A_a+A_b+\frac{\Lambda_z}{\Lambda_0}(A_c+A_d); \]

\[ A_p=-(A_a+A_b)+\frac{\Lambda_p}{\Lambda_0}(A_c+A_d), \tag{14} \]

where

\[ A_a=\frac{\Lambda_z\Lambda_p}{\Lambda_0}t_a' \]

\[ =-\frac{\Lambda}{k}\, \frac{(t_{ph}-t_{zh})Chkx-(t_{p0}-t_{z0})Chk(h-x)}{Shkh}; \tag{15} \]

\[ A_b=\frac{\Lambda_z\Lambda_p}{\Lambda_0}t_b' =\frac{\Lambda}{k}\,t_0\, \frac{Chk(h-x)-Chkx}{Shkx}; \tag{16} \]

\[ A_c=-\Lambda_0t_c' =\frac{1}{h}\left[\Lambda_p(t_{p0}-t_{ph})+\Lambda_z(t_{z0}-t_{zh})\right]; \tag{17} \]

\[ A_d=-\Lambda_0t_d'=q\left(x-\frac{h}{2}\right), \tag{18} \]

it is seen that the components \(A_a\) and \(A_c\) depend on the temperature of the end points, while \(A_b\) and \(A_d\) depend on the internal heat sources.

This fact is entirely natural, since the model of the tooth zone under consideration (Fig. 1) is a linear system; hence the possibility of applying the superposition method in this case also follows.

Thermal equivalent circuit

To calculate the temperature distribution in the tooth and slot, it is necessary to know the conditions of heat exchange between the tooth zone and the adjacent parts of the machine—the air gap and the yoke. Thus, the boundary conditions on the basis of which the temperature distribution is calculated must be determined in the course of a complete thermal calculation of the machine, which is most conveniently done with the aid of a thermal equivalent circuit. Since the model of the tooth zone under consideration is connected with the remaining parts of the machine by the heat flows \(A_{z0}\), \(A_{p0}\), \(A_{zh}\), and \(A_{ph}\), it must be represented in the equivalent circuit in the form of a four-terminal network.

The relations between the heat flows and the temperature potentials of the poles of the four-terminal network must coincide with the actual relations for the tooth zone, and the total output of its internal heat sources must be equal to the losses in the calculated part of the slot and tooth. However, it is not at all required that the distribution of temperatures and heat flows inside the four-terminal network reflect the actual picture in any way.

To determine the parameters of the thermal equivalent circuit, we take as a basis the relation between temperatures and heat flows at the extreme points of the tooth zone, \(x = 0\) and \(x = h\), obtained on the basis of expressions (15)...(18)

\[ A_{p0} = \frac{\lambda}{k}\, \frac{(t_{ph} - t_{zh}) - (t_{p0} - t_{z0})\,Chkh}{Shkh}, \tag{19} \]

\[ A_{zh} = \frac{\lambda}{k}\, \frac{(t_{ph} - t_{zh})\,Chkh - (t_{p0} - t_{z0})}{Shkh}, \tag{20} \]

\[ A_{b0} = -A_{bh} = \frac{\lambda}{k}\, t_0\, \frac{1 - Chkh}{Shkh}, \tag{21} \]

\[ A_{c0} = A_{ch} = \frac{1}{h}\,[\Lambda_p (t_{p0} - t_{ph}) + \Lambda_z (t_{z0} - t_{zh})], \tag{22} \]

\[ A_{d0} = -A_{dh} = -q\,\frac{h}{2}. \tag{23} \]

Substituting expressions (19)...(23) into (14), we obtain the system of equations of a four-terminal network with internal heat sources (Fig. 2)

\[ \begin{aligned} A_{z0}={}& -G_1(t_{p0}-t_{z0}) + G_2(t_{ph}-t_{zh}) \\ &+ G_3(t_{z0}-t_{zh}) + G_4(t_{p0}-t_{ph}) - Q_z,\\[4pt] A_{p0}={}& G_1(t_{p0}-t_{z0}) - G_2(t_{ph}-t_{zh}) \\ &+ G_4(t_{z0}-t_{zh}) + G_5(t_{p0}-t_{ph}) - Q_p,\\[4pt] A_{zh}={}& -G_2(t_{p0}-t_{z0}) + G_1(t_{ph}-t_{zh}) \\ &+ G_3(t_{z0}-t_{zh}) + G_4(t_{p0}-t_{ph}) + Q_z,\\[4pt] A_{ph}={}& G_2(t_{p0}-t_{z0}) - G_1(t_{ph}-t_{zh}) + \end{aligned} \tag{24} \]

\[ +G_4(t_{z0}-t_{zh})+G_5(t_{n0}-t_{nh})+Q_n . \]

Here,

\[ G_1=\frac{\lambda}{k}\,th\,kh; \]

\[ G_2=\frac{\lambda}{k}\,\frac{1}{Sh\,kh}; \]

\[ G_3=\frac{\lambda_z^2}{h\lambda_0}; \]

\[ G_4=\frac{\lambda_z\lambda_n}{h\lambda_0}; \]

\[ G_5=\frac{\lambda_n^2}{h\lambda_0}; \]

\[ Q_z=\frac{q\lambda_z h}{2\lambda_0}-t_0\frac{\lambda}{k}\,\frac{Ch\,kh-1}{Sh\,kh}; \tag{25} \]

\[ Q_n=\frac{q\lambda_n h}{2\lambda_0}+t_0\frac{\lambda}{k}\,\frac{Ch\,kh-1}{Sh\,kh}. \tag{26} \]

By transferring \(Q_z\) and \(Q_n\) in equations (24) to the left-hand side, we obtain a system of equations for a four-terminal network without internal heat sources, with new values of the heat fluxes:

Fig. 2. Representation of the tooth zone as a four-terminal network.

\[ Q_{z0}=A_{z0}+Q_z; \]

\[ Q_{n0}=A_{n0}+Q_n; \]

\[ Q_{zh}=A_{zh}-Q_z; \]

\[ Q_{nh}=A_{nh}-Q_n. \]

This transformation is equivalent to moving the internal distributed heat sources to the terminals of the four-terminal network and replacing them by concentrated sources with outputs \(Q_z\) and \(Q_n\), as shown in the diagram (Fig. 2).

Thus, it remains to select the corresponding passive four-terminal-network circuit and to determine the values of its resistances. Since the heat flows are related by

\[ Q_{z0}+Q_{p0}-Q_{zh}-Q_{ph}=0, \]

and the temperature differences by the relation

\[ (t_{p0}-t_{z0})-(t_{ph}-t_{zh})+(t_{z0}-t_{zh})-(t_{p0}-t_{ph})=0, \]

one of the equations can be eliminated.

\[ \begin{aligned} Q_{z0}={}&-(G_1-G_4)(t_{p0}-t_{z0})-(G_4-G_2)(t_{ph}-t_{zh}) \\ &+(G_3+G_4)(t_{z0}-t_{zh}); \\[4pt] Q_{zh}={}&(G_4-G_2)(t_{p0}-t_{z0})+(G_1-G_4)(t_{ph}-t_{zh}) \\ &+(G_3+G_4)(t_{z0}-t_{zh}); \\[4pt] Q_{p0}={}&(G_1+G_5)(t_{p0}-t_{z0})-(G_2+G_5)(t_{ph}-t_{zh}) \\ &+(G_4+G_5)(t_{z0}-t_{zh}). \end{aligned} \tag{27} \]

The system obtained has six coefficients; however, only four of them are independent, since

\[ G_1+G_5=(G_4+G_5)+(G_1-G_4), \]

\[ G_2+G_5=(G_4+G_5)-(G_4-G_2). \]

It follows from this that in the equivalent circuit there must be four independent resistances, which cannot be reduced to a smaller number when it is transformed. At the same time, it follows from equations (24) that the four-terminal-network circuit must be symmetric, since, when the indices \(0\) and \(h\) are interchanged and the signs before the heat flows are changed, the original system is obtained again. These requirements are satisfied, for example, by the circuit in Fig. 3.

Let us write the system of equations for this circuit and, comparing it with system (27), derive expressions for the resistances

\[ R_z=\frac{h}{2\Lambda_z}; \tag{28} \]

\[ R_p=\frac{h}{2\Lambda_p}; \tag{29} \]

\[ R_0=\frac{\lambda_0}{\Lambda_z\Lambda_p}\,\frac{\operatorname{Sh}kh}{k} -\frac{1}{\operatorname{Ch}kh+1-\dfrac{2}{kh}\operatorname{Sh}kh}; \tag{30} \]

\[ R=\frac{\lambda_0 h}{4\Lambda_z\Lambda_p}\, \frac{1}{\dfrac{\operatorname{Sh}kh}{kh}-1}. \tag{31} \]

The equivalent circuit obtained, being part of the complete thermal circuit of the machine, makes it possible to determine the temperature at the extreme points of the tooth zone, from which, if necessary, the temperature distribution over the height of the tooth and the slot can be calculated by means of formulas (9)...(13).

The real tooth zone, unlike the one considered, has a more complex configuration and a nonuniform filling of the slot because of

the presence of a wedge, interlayer spacers, etc., especially if there are several windings in the slot. Taking this into account, in the case of a significant difference one may divide the slot by height into two or more sections and construct the equivalent circuit in the form of the corresponding number of cascaded quadripoles. The parameters of each of them are calculated from the mean width of the slot and the tooth in the corresponding section. However, for the usual form of slots and one winding, calculations carried out with and without this refinement give results differing by no more than 5–10% of the maximum excess of the temperature in the slot over the adjacent parts of the machine.

On the other hand, in practical calculations one may, neglecting the temperature drop in the insulation of the slot bottom, take \(t_{nh}=t_{zh}\) and short-circuit the corresponding poles of the quadripole. In this case the equivalent circuit is simplified (Fig. 4).

The parameters of the simplified circuit are

\[ Q_h = Q_z + Q_p = \frac{qh}{2}; \]

\[ R_h = \frac{h}{\Lambda_0}\left(1 - \frac{Sh\,kh}{kh\,Ch\,kh}\right); \]

\[ R_{z0} = \frac{1}{\Lambda_z}\frac{Sh\,kh}{k\,Ch\,kh}; \]

\[ R_{p0} = \frac{1}{\Lambda_p}\frac{Sh\,kh}{k\,Ch\,kh}. \]

When calculation is made according to the simplified circuit, the value of \(t_{nh}\) is found to be underestimated; however, the temperature under the wedge \(t_{n0}\), whose value is the most important, practically does not change.

Temperature distribution over the width of the slot

To take into account the temperature distribution over the width of the slot, let us assume that the thermal energy released per unit volume per second, \(p_m\), is divided into two parts: the component \(p_x\), which is subsequently transmitted only along the \(x\)-axis, and the component \(p_y\), which forms the transverse flux.

Let, for \(x=\mathrm{const}\), \(p_y=\mathrm{const}\) as well. In this case the temperature over the width of the slot is distributed according to a parabola (Fig. 5).

Within the slot,

\[ t = t_{nm} - \frac{p_y y^2}{2\lambda_z}. \tag{32} \]

It follows from (32) that the maximum temperature is

\[ t_{nm} = t_n + \frac{p_y b^2}{2\lambda_z}. \tag{33} \]

Then the mean temperature for the given value of \(x\) is equal to

\[ t_n = \frac{1}{b}\int_0^b t\,dy = t_n + \frac{p_y b^2}{3\lambda_z}. \tag{34} \]

On the other hand,

Fig. 3. Equivalent four-terminal network diagram.

Fig. 4. Simplified thermal circuit.

\[ t_n = t_z + p_y b \frac{\Delta_n}{\lambda_n}, \tag{35} \]

where \(\Delta_n\) is the thickness of the slot insulation,
\(\lambda_n\) is its thermal conductivity,

\[ b = \frac{b_p}{2} - \Delta_n . \tag{36} \]

Taking into account that \(p_y b\) is the density of the heat flux through the side surface of the slot to one side,

\[ p_y b = \frac{a}{2}, \tag{37} \]

Figure labels: tooth; slot; \(t_z\); \(t_y\); \(t_m\); \(y\); \(O\); \(\Delta_n\); \(b\); \(b_p\).

Fig. 5. Temperature-distribution curve across the slot width for a random-wound winding.

On the basis of expressions (34), (35), and (37), we obtain

\[ t_n - t_z = \frac{a}{2} \left( \frac{\Delta_n}{\lambda_n} + \frac{b}{3\lambda_e} \right). \tag{38} \]

At the same time, according to (4),

\[ \frac{a}{t_n - t_z} = \Lambda, \tag{39} \]

whence

\[ \Lambda = \frac{2}{ \dfrac{\Delta_n}{\lambda_n} + \dfrac{b}{3\lambda_e} }. \tag{40} \]

thermal conductivity between the slot and the tooth, taking into account the temperature drop in the insulation of the conductors.

The temperature in the middle of the slot, according to (33) and (34), is

\[ t_{\mathrm{pm}} = t_{\mathrm{p}} + \frac{p_y b^2}{6\lambda_e}. \]

Taking (37) and (39) into account, we obtain

\[ t_{\mathrm{pm}} = t_{\mathrm{p}} + (t_{\mathrm{p}} - t_{\mathrm{z}})\frac{\Lambda b}{12\lambda_e}. \tag{41} \]

Using expression (9), we finally obtain

\[ \begin{aligned} t_{\mathrm{pm}} &= t_{\mathrm{p}} + (t_a + t_b)\frac{\Lambda b}{12\lambda_e} \\ &= \left(\frac{\Lambda_z}{\Lambda_p} + \frac{\Lambda b}{12\lambda_e}\right)(t_a + t_b) + t_c + t_d . \end{aligned} \tag{42} \]

Conclusion

1. The obtained equivalent thermal circuit of the tooth zone of an electric machine differs from the commonly used circuits in that it makes it possible to calculate not the average temperature of the winding or of the tooth, but the temperature at the most characteristic points: under the wedge, at the bottom of the slot, in the tooth head and at the tooth base; that is, to obtain a more concrete picture of the heating of the part of the machine most heavily loaded thermally.

2. The derived expressions for the temperature distribution in the slot and in the tooth make it possible to calculate the temperature of the hottest point, the average temperature of the winding, and to identify the factors affecting the nonuniformity of winding heating over the height of the slot.

3. Calculations carried out for various specific cases show that the principal causes of the nonuniform temperature distribution in the slot are the temperature difference over the height of the tooth and the heat-insulating action of the wedge. If the heat flow is directed from the tooth head to the base, as in the stators of enclosed machines, then the action of both factors is combined, and the maximum temperature, according to the calculation, occurs at the point under the wedge.

With the reverse direction of the flow, as, for example, in an armature without ventilation ducts, the temperature distribution is more uniform, and the hottest point usually lies near the middle of the slot. Thus, the calculated character of the temperature distribution agrees with practical data.

References

1. M. A. Mikheev. Fundamentals of Heat Transfer. GEI, 1956.
2. G. Gotter. Heating and Cooling of Electric Machines. GEI, 1961.
3. I. M. Postnikov. Design of Electric Machines. GITL UkrSSR, 1960.
4. M. A. Lyubchik. Calculation of the equivalent coefficient of thermal conductivity of windings of electrotechnical devices. Proceedings of Higher Educational Institutions, Electromechanics, No. 2, 1958.

Received by the editors in June 1962.