Determination of the Maximum Temperature Rise in Electric Machines by an Indirect Method

I. G. Kuleev

Determining the maximum heating temperature of powerful electric machines by the direct method presents great difficulties, owing to the fact that an expensive direct-current converter installation is required, by means of which the load is regulated. An indirect method is proposed for determining the maximum heating temperature of a machine, by which the cost of the test installation can be considerably reduced while retaining a relatively high accuracy of the method. According to this method, the maximum temperature rise of the machine is determined experimentally only at an insignificant load. At nominal load it can be calculated from the formula

\[ \tau_{n\,max} = \tau_{max}\,\frac{P_n}{P}, \tag{1} \]

where \(\tau_{max}\) is the maximum temperature rise during the test,
\(\tau_{n\,max}\) is the maximum temperature rise at normal load,
\(P_n\) is the nominal power,
\(P\) is the power of the machine during the test.

The correctness of equality (1) follows directly from comparison of the equations of the thermal state of the electric machine for the first and second cases of loading.

Let, for a load less than nominal, this equation be

\[ P\,dt = c\,d\tau + A\tau\,dt, \]

where \(c\) is the heat capacity of the machine,
\(A\) is the heat transfer,
\(\tau\) is the temperature rise of the machine above the temperature of the ambient air,
\(t\) is time.

The solution of this equation has the form:

\[ \tau = \tau_{max}\left(1 - e^{-\frac{A}{c}t}\right). \]

For nominal load it has the form:

\[ P_n\,dt = c\,d\tau_n + A\tau_n\,dt, \]

and its solution is

\[ \tau_n = \tau_{n\,max}\left(1 - e^{-\frac{A}{c}t}\right). \]

Under steady-state conditions, when all the heat released in the machine is given off to the surrounding medium (Fig. 1), the equation of the thermal state of an electric machine takes the form, for the first case, \(A\tau_{max}=P\), and for the second case, \(\tau_{n\,max}A=P_n\); hence

\[ \frac{\tau_{n\,max}}{\tau_{max}}=\frac{P_n}{P}. \]

Consequently, the maximum temperature rise at rated load is

\[ \tau_{n\,max}=\tau_{max}\frac{P_n}{P}. \]

It follows from this formula that the ratio of the maximum heating temperatures of an electric machine is proportional to the ratio of the losses produced in it. Consequently, it is sufficient to bring the machine to the steady-state thermal condition \(\tau_{max}\) at any load less than normal, in order to use it to find the maximum heating temperature at rated load. The results obtained are sufficiently accurate, since no assumptions whatever were made in deriving this formula. As for the cooling and heating conditions of the machine, they were exactly the same in both the first and the second case. Experiments carried out at Plant 653 on the heating of the AD—32—4 motor confirm the determination of the maximum heating temperature at rated power from the maximum temperature found experimentally at a load less than rated.

Fig. 1

Fig. 2

According to the factory data, at a load

\[ P=\frac{1}{2}P_n \]

the maximum temperature rise is \(\tau_{max}=22^\circ\text{C}\). The losses in the machine in this case are \(P=268\) W. At rated load \(P_n=3.2\) kW, the maximum temperature rise is \(\tau_{n\,max}=57^\circ\text{C}\). The losses in the machine are \(P_n=628\) W,

\[ \tau_{n\,max}=\tau_{max}\frac{P_n}{P}=52^\circ, \]

that is, the discrepancy is \(5^\circ\), which amounts to \(8.5\%\).

The test was carried out according to the following arrangement (Fig. 2): here AD is the motor under test, G is a direct-current generator, M is a direct-current motor, and AG is an asynchronous generator.