ESTIMATION OF THE TEMPERATURE DISTRIBUTION IN THE WINDING UNDER NONUNIFORM COOLING ALONG ITS LENGTH

V. G. Ryazanov

(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)

The principal method for the experimental determination of the thermal state of electrical machines with a power of up to 5000 kW is the measurement of the average resistance of windings having terminals.

In powerful electrical machines having intensive ventilation of the active parts, and especially in high-speed machines with relatively long end parts, there may be significant nonuniformity of winding heating along the length of the turn. Since in high-voltage machines the placement of thermocouples directly on the active copper is associated with considerable technological and design difficulties, it is usually preferable to carry out sufficiently detailed measurements of the temperature rises of the tooth zone of the active steel by means of thermocouples embedded in the teeth, and then, by adding to them the calculated temperature drops in the winding insulation, to determine the thermal state of the winding in the slot part. Subsequent comparison of the measured average temperature rise of the winding with the temperature rise of the slot part makes it possible to determine the average temperature rise of the end parts.

It often turns out that the average levels of temperature rise of the slot and end parts differ substantially; therefore, when the average temperature rise of the winding is close to the maximum permissible for the given insulation class, the reliability of operation of the insulation in the more heated section of the winding becomes highly questionable. In this case it becomes necessary to represent the distribution of the copper temperature rises more accurately, taking into account the axial heat flows in the winding that equalize the temperature rises of sections with different cooling intensity. Let us consider a method for calculating winding temperature rises with allowance for axial heat flows when processing measurement results.

The calculation is simpler for a symmetric axial ventilation system. Let us distinguish two sections of the winding: from the middle of the core to the end face, and from the end face to the end heads (Fig. 1).

The Poisson equation for each section is:

\[ \lambda f \frac{d^2 \vartheta}{dx^2} - (\vartheta - \vartheta_c) A = -p, \tag{1} \]

where \(\lambda\) and \(f\) are the coefficient of thermal conductivity and the cross section of the copper,

\(\vartheta\) is the temperature rise of the winding copper above the incoming air,

\(\vartheta_c\) is the temperature rise of the medium surrounding the winding (core steel or air) above the incoming air,

\(\lambda\) is the conductance for heat transfer to the surrounding medium per unit length of the winding, \(p\) is the copper loss per unit length of the winding.

The boundary conditions for each section are:

\[ -\lambda f \left.\frac{d\vartheta}{dx}\right|_{x=l}=0 \tag{2} \]

\[ -\lambda f \left.\frac{d\vartheta}{dx}\right|_{x=0}=\mp q. \tag{3} \]

In (3), \(q\) is the axial heat flux (taken as positive toward the end-winding part), and here and below the upper sign refers to the slot section.

The solution of (1), taking (2) and (3) into account, is

\[ \vartheta=\vartheta_c+\frac{p}{\Lambda}\mp q\frac{\operatorname{ch}(l-x)m}{\lambda f m\cdot \operatorname{sh} ml}, \qquad \text{where } m=\sqrt{\Lambda/\lambda f}. \tag{4} \]

From (4), the average copper temperature rise is

\[ \vartheta_{\mathrm{av}.}=\frac{1}{l}\int_0^l \vartheta dx =\vartheta_c+\frac{1}{\Lambda}\left(p\mp \frac{q}{l}\right) \tag{5} \]

and the copper temperature rise at the core end is

\[ \vartheta_T=\vartheta_c+\frac{p}{\Lambda}\mp q\frac{1}{\lambda f m\,\operatorname{th} ml}. \tag{6} \]

Let us assign the indices 1 and 2, respectively, to the end-winding and slot parts. Then, taking into account the obvious equality

\[ \vartheta_{T1}=\vartheta_{T2} \tag{7} \]

it follows from expressions (6) that

\[ q=\lambda f \frac{(\vartheta_{c2}-\vartheta_{c1})+\left(\dfrac{p_2}{\Lambda_2}-\dfrac{p_1}{\Lambda_1}\right)} {\dfrac{1}{m_1\operatorname{th} m_1l_1}+\dfrac{1}{m_2\operatorname{th} m_2l_2}}. \tag{8} \]

However, we also have at our disposal the measured value of the average winding temperature rise,

\[ \vartheta_{\mathrm{av.\ wind.}}= \frac{\vartheta_{\mathrm{av}.1}\cdot l_1+\vartheta_{\mathrm{av}.2}\cdot l_2}{l_1+l_2}, \tag{9} \]

from which, taking (5) into account, \(q\) can also be determined:

\[ q=\frac{\Lambda_1\cdot \Lambda_2}{\Lambda_2-\Lambda_1} \left[ \vartheta_{\mathrm{av.\ wind.}}(l_1+l_2) -l_1\left(\frac{p_1}{\Lambda_1}+\vartheta_{c1}\right) -l_2\left(\frac{p_2}{\Lambda_2}+\vartheta_{c2}\right) \right]. \tag{10} \]

The disadvantage of each of expressions (8) and (10), taken separately, is the circumstance that, along with the measured quantities \(\vartheta_{c1}\) and \(\vartheta_{c2}\) and the losses \(p_1\) and \(p_2\), readily calculated from the measured winding resistance and current, we use the calculated quantities \(\Lambda_1\) and \(\Lambda_2\). Moreover, if the value \(\Lambda_2\) is known sufficiently well, a reliable calculation of \(\Lambda_1\) is very difficult because of the uncertainty of the heat-transfer coefficient from the surface of the coils and of the very magnitude of their heat-emitting surface (presence of spacers, ties, bands). Therefore, the simultaneous solution of (8) and (10) with respect to \(q\) and \(\Lambda\) makes it possible to determi-

divide not only \(q\), and consequently the distribution of copper overheating according to (4), but also the value \(\Lambda_1\), which in itself is quite valuable.

If it is necessary to increase the accuracy of the processing, the values \(p_1\) and \(p_2\) can be refined from the average overheatings of the sections obtained from (5), and the calculation repeated.

Let us give an example of processing experimental data. Motor of type A3-500/6000, asynchronous, \(500\ \text{kW}\), \(6\ \text{kV}\), \(57\ \text{A}\), \(3000\ \text{rpm}\). Parameters in the calculation for one half of the machine, for one slot: \(l_1 = 0.439\ \text{m}\), \(l_2 = 0.23\ \text{m}\), \(f = 3.28 \cdot 10^{-4}\ \text{m}^2\), \(\Lambda_2 = 6.55\ \text{W}/\text{m}^{\circ}\text{C}\). The experimental data and the results of calculations of \(q\) and \(\Lambda_1\) are in Table 1.

Table 1

In Table 1 the values of the specific losses are determined at the average overheating of the winding \(\vartheta_{\text{av.winding}}\). In the second line of regime 4 they have been refined according to the results of the first calculation. The value \(p_2\) includes additional losses in the copper.

The results of calculating the copper overheatings are presented in Fig. 1 for regime 4.

From the example given, some conclusions may be drawn:

1. The magnitude of the axial heat flux in the rated regime amounts to 32% of the losses in the slot part, i.e., with the calculated temperature drop in the insulation of \(23.5^{\circ}\text{C}\) \(\left(= \dfrac{p_2}{l_2}\right)\), the axial flux lowers the average temperature of the slot part by \(7.5^{\circ}\text{C}\), while the maximum temperature is lower by \(5.5^{\circ}\text{C}\).

2. From the four regimes an average value \(\lambda_1 = 3.89\ \dfrac{\text{W}}{\text{m}^{\circ}\text{C}}\) was obtained, with a relative error of up to \(\pm 6\%\), determined mainly by the accuracy of the measurements. Indeed, in regime 4 a higher copper overheating was measured than in regime 3, despite the smaller values of the inlet-air temperature and current.

3. Refining the value \(p\) according to the results of the first calculation had practically no effect on the values of the overheatings and \(\lambda_1\). The method described is applicable to stator or rotor windings, including machines with other ventilation systems. For example, with a symmetrical ra-

![Figure 1]

Fig. 1. — — — calculation by (4), — — — calculation by (4) and (9) without taking \(q\) into account, \((q=0)\), ✦ measured tooth temperature rises,
\(1\) — calculation of the first approximation, \(2\) — calculation of the second approximation

For a radial ventilation system one may use the expressions given above if one sets:

\[ \Lambda_2=\Lambda_{\mathrm{p}}\cdot(1-\beta)+\Lambda_{\mathrm{k}}\cdot\beta, \tag{11} \]

\[ \vartheta_{c2}= \frac{ \vartheta_{\mathrm{p}}\cdot\Lambda_{\mathrm{p}}\cdot(1-\beta) + \vartheta_{\mathrm{k}}\cdot\Lambda_{\mathrm{k}}\cdot\beta }{ \Lambda_{\mathrm{p}}\cdot(1-\beta)+\Lambda_{\mathrm{k}}\cdot\beta }, \tag{12} \]

where

\[ \begin{aligned} \vartheta_{\mathrm{p}} \quad &—\ \text{measured average temperature rise of the steel along the core length,}\\ \vartheta_{\mathrm{k}} \quad &—\ \text{measured average temperature rise of the air in the winding region in the radial ducts,}\\ \Lambda_{\mathrm{p}}\ \text{and}\ \Lambda_{\mathrm{k}} \quad &—\ \text{calculated specific conductances of the winding, respectively in the package section and in the radial duct,}\\ &\quad \text{again per unit length of the winding,}\\ \beta=\frac{b_{\mathrm{k}}}{b_{\mathrm{p}}+b_{\mathrm{k}}} \quad &—\ \text{relative width of the radial duct.} \end{aligned} \]