THERMAL CALCULATION OF ELECTRIC MOTORS UNDER RANDOM LOAD

Yu. M. Bashagurov, D. I. Sannikov

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

The known classical methods for checking electric motors for heating are based, as a rule, on representing the variation of the static load by deterministic functions of time. However, first, cases of deterministic load diagrams constitute only a certain part of the diversity of operating modes of electric drives, among which modes with a random character of load variation over time are widespread. Second, representing the operating load of electric motors by deterministic functions is in most cases a certain idealization of the actual load graph.

It is practically impossible to characterize a random load by a deterministic function; therefore, this article proposes a method for calculating the random thermal regime of asynchronous electric motors, based on the known principles of the theory of random functions [1].

The asynchronous motor is proposed to be regarded as a one-dimensional stationary linear dynamic system that transforms the random current function \(I(t)\) into the random temperature function \(\Theta(t)\), where the former is the input and the latter the output of the dynamic system under consideration.

Obtaining the law of temperature distribution by calculation requires knowledge of the multidimensional distribution laws of the input function \(I(t)\), which proves too complicated for engineering applications, since in practice information about the input process is usually limited.

Therefore, as is done in most practical applications, we shall use the correlation theory of random processes, which is based on two principal characteristics of a random function: the mathematical expectation \(m(\Theta)\) and the variance \(D(\Theta)\) of the output process.

As the operator of the dynamic system modeling the heating process of the winding of a closed-construction asynchronous motor, a system of three differential heating equations is adopted when the motor is considered as a thermal system of three bodies.

The division of an asynchronous motor into three bodies—the stator winding (1), the squirrel-cage rotor (2), and the stator core with the frame (3)—is the most natural from the point of view of the mutual influence of the individual parts of the motor during the heating process. Increasing the number of bodies

more than three, as studies show, no longer gives any substantial refinement, but considerably complicates the calculation.

The system of differential equations for the heating of three bodies is:

\[ \begin{aligned} C_1 \frac{d\Theta_1}{dt} + \Theta_1 \Lambda_1 - \Theta_3 \Lambda_1 &= P_1;\\ C_2 \frac{d\Theta_2}{dt} + \Theta_2 \Lambda_2 - \Theta_3 \Lambda_2 &= P_2;\\ C_3 \frac{d\Theta_3}{dt} + \Theta_3(\Lambda_1+\Lambda_2+\Lambda_3) - \Theta_1\Lambda_1 - \Theta_2\Lambda_2 &= P_3, \end{aligned} \tag{1} \]

where \(P_1\) is the losses in the stator winding;
\(P_2\) is the losses in the rotor winding and additional losses;
\(P_3\) is the losses in the steel and mechanical losses;
\(\Theta_1, \Theta_2, \Theta_3\) are the temperature rises above the ambient air, respectively for the first, second, and third bodies;
\(\Lambda_1\) is the thermal conductance from the stator winding to the stator core;
\(\Lambda_2\) is the total thermal conductance from the rotor to the stator through the gap and from the rotor through the air to the frame;
\(\Lambda_3\) is the thermal conductance from the stator core through the frame to the ambient air;
\(C_1\) is the heat capacity of the stator winding;
\(C_2\) is the total heat capacity of the rotor;
\(C_3\) is the heat capacity of the stator core, the frame, and the end shields.

Let us determine the dependence of the losses on the load:

\[ \begin{aligned} P_1 &= P_{10} + P_{\mathrm{ln}} \cdot K^2;\\ P_2 &= P_{\mathrm{add}} + P_{\mathrm{m2}} \cdot K^2;\\ P_3 &= P_{\mathrm{st}} + P_{\mathrm{mech}}, \end{aligned} \tag{2} \]

where \(K = I_n/I_{nn}\);
\(I_n\) is the load current of the stator winding;
\(I_{nn}\) is the load current of the stator winding in the rated operating mode;
\(P_{10}\) is the losses in the stator winding at no load;
\(P_{\mathrm{ln}}\) is the losses in the stator winding due to load current in the rated operating mode.

The losses in the windings may be taken at their mean temperature (or at the permissible temperature rise).

According to the principle of superposition, the temperature rises will consist of two components, due to constant and variable losses:

\[ \begin{aligned} \Theta_1 &= \Theta_{1c} + \Delta\Theta_1(t);\\ \Theta_2 &= \Theta_{2c} + \Delta\Theta_2(t);\\ \Theta_3 &= \Theta_{3c} + \Delta\Theta_3(t). \end{aligned} \tag{3} \]

For a sufficiently long operating time, the temperature rises from the constant components reach steady-state values. These steady-state values are equal to:

\[ \begin{aligned} \Theta_{1c} &= \frac{P_{10}}{\Lambda_1} + \frac{P_0}{\Lambda_3};\\ \Theta_{2c} &= \frac{P_{\mathrm{add}}}{\Lambda_2} + \frac{P_0}{\Lambda_3}; \end{aligned} \tag{4} \]

\[ \Theta_{3c}=\frac{P_0}{\Lambda_3}, \]

where

\[ P_0=P_{10}+P_{\mathrm{add}}+P_{\mathrm{st}}+P_{\mathrm{mech}}. \]

The components \(\Delta\Theta_1(t)\), \(\Delta\Theta_2(t)\), \(\Delta\Theta_3(t)\) depend on the law of variation of the load \(K^2(t)\) and can be obtained from (1) by substituting:

\[ \begin{aligned} P_1&=P_{\mathrm{pn}}\cdot K^2;\\ P_2&=P_{\mathrm{m2}}\cdot K^2;\\ P_3&=0. \end{aligned} \tag{5} \]

Then we obtain

\[ \begin{aligned} C_1\frac{d\Delta\Theta_1}{dt}+\Delta\Theta_1\Lambda_1-\Delta\Theta_3\Lambda_1&=P_{\mathrm{pn}}\cdot K^2;\\ C_2\frac{d\Delta\Theta_2}{dt}+\Delta\Theta_2\Lambda_2-\Delta\Theta_3\Lambda_2&=P_{\mathrm{m2}}\cdot K^2;\\ C_3\frac{d\Delta\Theta_3}{dt}+\Delta\Theta_3(\Lambda_1+\Lambda_2+\Lambda_3)-\Delta\Theta_1\Lambda_1-\Delta\Theta_2\Lambda_2&=0. \end{aligned} \tag{6} \]

System (6) is the operator of the dynamic system under consideration.

In the given problem it is necessary to determine the mean value of the temperature rise of the winding \(\overline{\Theta}_1\) and the variance \(D(\Delta\Theta_1)\).

Let us pass from the system of equations (6) to one equation with respect to \(\Delta\Theta_1\). Denote \(\Lambda_c=(\Lambda_1+\Lambda_2+\Lambda_3)\) and \(K^2=y\).

Differentiating the third equation of the system once, and the second twice, and carrying out the corresponding substitution operations, we obtain the differential equation with respect to \(\Delta\Theta_1\):

\[ \frac{d^3\Delta\Theta_1}{dt^3} +a_1\frac{d^2\Delta\Theta_1}{dt^2} +a_2\frac{d\Delta\Theta_1}{dt} +a_3\Delta\Theta_1 = b_0\frac{d^2y}{dt^2} +b_1\frac{dy}{dt} +b_2y, \tag{7} \]

where

\[ a_1= \frac{ \frac{C_2\cdot C_3}{\Lambda_2} +\frac{\Lambda_c\cdot C_1\cdot C_2}{\Lambda_1\cdot\Lambda_2} +\frac{C_1\cdot C_3}{\Lambda_1} }{n}; \]

\[ a_2= \frac{ \frac{C_2(\Lambda_2+\Lambda_3)}{\Lambda_2} +C_3 +\frac{C_1(\Lambda_1+\Lambda_3)}{\Lambda_1} }{n}; \]

\[ a_3=\frac{\Lambda_3}{n}; \qquad b_0=\frac{P_{\mathrm{pn}}}{C_1}; \]

\[ b_1= \frac{ \frac{P_{\mathrm{pn}}}{\Lambda_1} \left( \frac{\Lambda_c\cdot C_2}{\Lambda_2}+C_3 \right) }{n}; \]

\[ b_2= \frac{ P_{\mathrm{m2}}+ P_{\mathrm{pn}} \left( \frac{\Lambda_1+\Lambda_3}{\Lambda_1} \right) }{n}; \]

\[ n=\frac{C_1\cdot C_2\cdot C_3}{\Lambda_1\cdot\Lambda_2}. \]

The equation for the mean value of the temperature rise follows from (7) by averaging over the ensemble of both its parts:

\[ \frac{d^3 \overline{\Delta \Theta_1}}{dt^3} + a_1 \frac{d^2 \overline{\Delta \Theta_1}}{dt^2} + a_2 \frac{d \overline{\Delta \Theta_1}}{dt} + a_3 \overline{\Delta \Theta_1} = b_0 \frac{d^2 \overline{y}}{dt^2} + b_1 \frac{d \overline{y}}{dt} + b_2 \overline{y}. \tag{8} \]

From the condition of stationarity of the process \(I^2(t)\), the mathematical expectation \(y=\mathrm{const}\); therefore (8) is rewritten as

\[ \frac{d^3 \overline{\Delta \Theta_1}}{dt^3} + a_1 \frac{d^2 \overline{\Delta \Theta_1}}{dt^2} + a_2 \frac{d \overline{\Delta \Theta_1}}{dt} + a_3 \overline{\Delta \Theta_1} = b_2 \overline{y}. \tag{9} \]

The system described by equation (9) is stable. As is known, in this case the solution contains a constant term and a sum of decaying exponentials, which, after a sufficiently long time has elapsed, may be neglected because of their smallness. Thus, to determine the mean value \(\overline{\Delta \Theta_1}\) from equation (9), it is sufficient to set \(\overline{\Delta \Theta_1}=\mathrm{const}\); then

\[ a_3 \overline{\Delta \Theta_1} = b_2 \overline{y}. \tag{10} \]

Whence it follows that

\[ \overline{\Delta \Theta_1} = \frac{b_2}{a_3}\,\overline{y} = \left( \frac{P_{\mathrm{n}} + P_{\mathrm{M}2}}{A_3} + \frac{P_{\mathrm{n}}}{A_1} \right)\cdot \overline{y}. \tag{11} \]

The total mean temperature rise of the winding above the ambient temperature, according to expressions (3), (4), and (11), has the form

\[ \overline{\Theta_1} = \left( \frac{P_{10}}{A_1} + \frac{P_0}{A_3} \right) + \left( \frac{P_{\mathrm{n}} + P_{\mathrm{M}2}}{A_3} + \frac{P_{\mathrm{n}}}{A_1} \right)\cdot \overline{y}. \tag{12} \]

To determine the variance \(D(\Theta_1)\), it is necessary to know the correlation function of the process \(y(t)\). In our case, the normalized correlation function of the process \(y(t)\) has the form

\[ R_y(\tau) = e^{-\alpha \tau}. \tag{13} \]

The spectral density of the temperature rise (1) is

\[ S_{\theta_1}(\omega) = S_y(\omega)\,|\Phi(j\omega)|^2 \cdot I_{\mathrm{n}}^4, \tag{14} \]

where \(S_y(\omega)\) is the spectral density of the input process;
\(|\Phi(j\omega)|^2\) is the square of the modulus of the temperature frequency characteristic of the motor.

The frequency characteristic was obtained by applying to the motor input a harmonic process of the square of the current. The ratio of the amplitudes of the processes at the output and input of the system, as is known, gives the amplitude frequency characteristic.

The square of the modulus of the frequency characteristic is approximated by the expression

\[ |\Phi(j\omega)|^2 = \Phi_0 e^{-\gamma \omega} = 0.242 e^{-2250\cdot \omega}, \tag{15} \]

where \(\Phi_0\) is the value of the square of the modulus of the frequency characteristic at \(\omega=0\).

The spectral density of the process \(y(t)\), corresponding to the correlation function (13), has the form

\[ S_y(\omega)=\frac{\alpha}{\pi}\,\frac{D(y)}{\alpha^2+\omega^2}. \tag{16} \]

The variance of the temperature rise

\[ D(\Theta_1)=2\int\limits_0^\infty S_{\Theta_1}(\omega)\,d\omega = \frac{2\cdot I_{1\text{н}}^{4}D(y)\Phi_0}{\pi} \int\limits_0^\infty \frac{e^{-\gamma\omega}}{a^2+\omega^2}\,d\omega = \]

\[ = \frac{2D(I_1^2)\Phi_0}{\pi} \left[\operatorname{ci}(a\gamma)\cdot\sin(a\gamma)-\operatorname{si}(a\gamma)\cdot\cos(a\gamma)\right]. \tag{17} \]

As an example of the application of the method set forth, below are calculations of the mean values and variances of the temperature rise of the stator winding of a 4A112M4 asynchronous motor.

At the input of the electric motor, by the method indicated in (2), realizations of the process \(J(t)\), simulated on a digital computer, were supplied, with a duration of 7 hours and with correlation functions of the form (13). The results of the experiment, for the indicated parameters of the load process and the frequency characteristic of the motor, are presented in the table.

Table

The results of the investigations show that the calculated data in determining the mean values and variances of the temperature rise of the stator winding differ from the experimental data by no more than 10 percent. Such a discrepancy may be considered satisfactory.

Conclusions

1. The application of the theory of random processes in thermal calculations of electric motors makes it possible to overcome the limitations of existing methods, which are intended mainly for deterministic load diagrams.

2. To determine the winding temperature rise in random modes, it is advisable to consider the motor as a dynamic system. The method presented is universal, since it can be applied to any modes, and is the most natural from the point of view of the physical representation of thermal processes in electric motors.

3. The proposed method of thermal calculation has been considered using the example of an asynchronous motor. The basic provisions of the method can also be extended to other types of electrical machines.

References

1. A. A. Sveshnikov. Applied Methods of the Theory of Random Functions. Moscow, “Nauka,” 1969.

2. Yu. M. Bashagurov, D. I. Sannikov. Investigation of the heating of electric motors in random modes. Izvestiya TPI, vol. 284, Tomsk, 1974.

3. N. P. Buslenko. Mathematical Modeling of Production Processes. Moscow, “Nauka,” 1964.