Proceedings of the Tomsk Polytechnic Institute Named after S. M. Kirov, Holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Vol. 284　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

RESPONSE SPEED WHEN SWITCHING OFF ELECTRIC MOTORS WITH BUILT-IN BRAKE ELECTROMAGNETS

V. S. GRINBERG, E. M. GUSELNIKOV, E. V. KONONENKO

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

Reducing the stopping time of mechanisms leads to a reduction in unproductive working time and to an increase in the efficiency of equipment utilization. In this connection, asynchronous electric motors with built-in spring brakes are becoming widespread; these make it possible to considerably reduce the stopping time and to improve operating safety.

To release the brake on the shaft of an electric motor when it is connected to the mains, brake electromagnets are used, the supply to which is provided from the stator winding of the electric motor through a rectifier (Fig. 1). When the electric motor is connected to the mains, voltage is applied through the rectifier to the electromagnet coil; the electromagnet operates, releasing the shaft of the electric motor. When the electric motor is disconnected from the mains, the armature of the electromagnet is released; the braking device operates, braking the shaft of the electric motor. The input of the rectifier (Fig. 1) is usually connected to the terminals of the stator winding directly (dashed line) or through additional contacts of the starter [4 and others].

Fig. 1. Control circuit of a brake electric motor

As experimental investigations have shown, the decisive role in the response speed of brake electric motors during switching off is played by the value of the electromagnet release time \(t_{\text{rel}}\); therefore, when designing brake electric motors it is necessary to take this value into account.

When the rectifier input is connected to the stator winding through additional contacts of the starter, the transient processes in the asynchronous electric motor and in the electromagnet proceed independently of one another. They have been studied quite fully [1—3 and others]. With a slow decrease in the speed of the electric motor after it is disconnected from the mains, the magnetic flux and the amplitude of the emf induced in the stator winding decrease according to an exponential law with the rotor time constant. In this case the release time of the electromagnet is determined by the known expression (provided that the valves are ideal):

\[ t_{\mathrm{rel}1}=T\cdot \ln \frac{i_{\mathrm{obm}}}{i_{\mathrm{rel}}}, \tag{1} \]

where

\[ T=\frac{L}{R} \]

is the time constant of the electromagnet with the armature attracted;

\(i_{\mathrm{obm}}\) is the current in the electromagnet winding at the instant of switching off;

\(i_{\mathrm{rel}}\) is the current at which the electromagnet ceases to hold its armature;

\(L\) and \(R\) are, respectively, the inductance and active resistance of the electromagnet.

In the case of direct connection of the rectifier input to the terminals of the stator winding, the transient processes in the electric motor and in the electromagnet exert a mutual influence on one another, since the stator winding of the electric motor remains connected with the electromagnet winding (electrical coupling through the rectifier) and with the rotor winding (magnetic coupling).

The presence of current in the stator winding leads to a slowing of the decrease in the magnetic flux and in the amplitude of the e.m.f. of the electric motor, as compared with the case of complete de-energization of the stator when switched off. This leads to a slow decrease in the voltage and current of the electromagnet, which in turn contributes to damping of the decay of the magnetic flux of the electric motor, etc. In this case the release time of the electromagnet \(t_{\mathrm{rel}2}\) increases considerably in comparison with the time \(t_{\mathrm{rel}1}\) and may exceed the permissible value \(t_{\mathrm{perm}}\).

The value of the release time \(t_{\mathrm{rel}2}\) can be determined simply if the law of variation with time of the electromagnet current after switching off is known. However, finding this law in analytical form without additional simplifications is not possible because of the presence of a nonlinear element (the rectifier) and the asymmetrical connection of the stator circuit (when the rectifier circuit is replaced by an equivalent linear load and the differential equations are transformed to orthogonal axes fixed relative to the stator, it is necessary to find the roots of a third-degree characteristic equation, which have cumbersome expressions).

For engineering calculations, one may use the simplified method proposed below for determining the law of variation of the electromagnet current and the magnitude of the release time \(t_{\mathrm{rel}2}\).

When deriving the law of variation of the electromagnet current during switching off, as a first approximation we assume that the transient processes in the stator circuit do not affect the transient processes in the rotor winding. In accordance with the adopted assumption, the amplitude of the e.m.f. induced in the stator winding and the mean value of the voltage across the electromagnet winding must decrease after switching off according to an exponential law with the rotor time constant. Under such consideration, the influence of the transient processes in the rotor winding on the transient processes in the electromagnet is not fully taken into account. However, the mutual influence of the transient processes in the electric motor and in the electromagnet can be accounted for more fully by substituting the corresponding parameters into the calculation expressions derived under the stated assumption.

In accordance with the foregoing, the voltage-balance equation of the electromagnet winding with respect to the mean values of the variables can be written in the form

\[ U_0 \cdot e^{-\frac{t}{T_2}} = R \cdot i + L \cdot \frac{di}{dt} \tag{2a} \]

or

\[ i_0 \cdot e^{-\frac{t}{T_2}} = i + T \cdot \frac{di}{dt}, \tag{2b} \]

where \(U_0\) and \(i_0\) are the average components of the voltage and current of the electromagnet in the rated operating mode before switching off;

\(i\) is the average component of the electromagnet current after switching off;

\[ T_2=\frac{x_0+x_2'}{2\cdot \pi \cdot f \cdot r_2'} \]
is the time constant of the rotor winding referred to the stator;

\(x_0\), \(x_2'\), and \(r_2'\) are, respectively, the mutual-induction reactance between stator and rotor, the leakage reactance, and the active resistance of the rotor;

\(f\) is the line frequency.

Solving equation (2b) with respect to the current \(i\), we obtain

\[ i=i_0 \cdot \frac{T_2 \cdot e^{-\frac{t}{T_2}} - T \cdot e^{-\frac{t}{T}}}{T_2-T}, \quad \text{if } T_2 \ne T, \tag{3a} \]

\[ i=i_0 \cdot \left(1+\frac{t}{T}\right)\cdot e^{-\frac{t}{T}}, \quad \text{if } T_2=T. \tag{3b} \]

Knowing the value of the current \(i_{\text{rel}}\), with the aid of expressions (3a) or (3b) it is not difficult to determine the value of the release time \(t_{\text{rel2}}\) by the method of successive approximations.

In accordance with the assumptions adopted, the calculated values of \(t_{\text{rel2}}\), determined from expressions (3), must obviously be smaller than the real values (experimental studies carried out on several prototype samples of brake motors showed that the values of the release time determined from oscillograms exceed the calculated values by a factor of 2 or more). Therefore, if the calculated values of \(t_{\text{rel2}}\) determined from these expressions exceed the permissible value of the release time \(t_{\text{perm}}\), then, when the brake motor is switched off according to the indicated circuit, the required response speed cannot be achieved.

If, however, these calculated values turn out to be less than the permissible value, then the question of the circuit’s response speed remains open: a more complete account is required of the mutual influence of the transient processes in the motor and in the electromagnet. The latter is conveniently accomplished by increasing the time constants used in the calculations. Good agreement of results can be obtained by replacing the quantities \(T_2\) and \(T\) by \(T_{\text{sum}}\) in expressions (2b) or (3b):

\[ \frac{i_{\text{rel}}}{i_0}=\left(1+\frac{t_{\text{rel2}}}{T_{\text{sum}}}\right)\cdot e^{-\frac{t_{\text{rel2}}}{T_{\text{sum}}}}, \tag{4} \]

where

\[ T_{\text{sum}}=T+T_2. \]

Comparison of the release-time values calculated from expression (4) with the experimental values showed that the discrepancy between the results usually does not exceed a few percent.

REFERENCES

1. K. P. Kovach, I. Rats. Transient Processes in Alternating-Current Machines. Moscow–Leningrad, Gosenergoizdat, 1963.

2. M. M. Sokolov et al. Electromagnetic Transients in an Asynchronous Electric Drive. Moscow, “Energiya,” 1967.

3. A. G. Slivanskaya, A. V. Gordon. Electromagnets with Built-in Rectifiers. Moscow, “Energiya,” 1970.

4. W. Baumann. Bauarten und Eigenschaften electromagnetisch gelüfteter Bremsen an Stoppmotoren, Konstruktion, 21 (1969), Heft 4.