STUDY OF ELECTROMECHANICAL TRANSIENT PROCESSES IN A SELF-BRAKING ELECTRIC MOTOR WITH AN ELECTROMAGNETIC INSERT

L. K. BURULKO, E. M. GUSELNIKOV, E. V. KONONENKO, T. V. CHESHEVA

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

Increasing labor productivity in industry, construction, agriculture, and other branches of the national economy poses to electrical-machine engineering the task of creating improved designs of electric motors which, by increasing speed and reducing the unproductive time spent on stopping them, must raise the productivity of actuating mechanisms.

One solution to this problem is the creation of electric motors with built-in and attached braking devices that ensure the stopping of the machine at the required moment.

At present, both abroad and in domestic industry, electric motors are produced mainly with built-in electromagnetic brakes for direct and alternating current. However, these electric motors do not satisfy all the requirements imposed on ideal electric motors with braking devices. Therefore it is considered [2] that such requirements can be most fully met by means of so-called self-braking electric motors (SBEM).

Unlike electric motors with built-in and attached electromagnetic brakes, in which a special winding is used to control the braking device, in self-braking electric motors the brake is controlled by means of the magnetic flux of the electric motor itself.

This paper considers and investigates one design of a self-braking electric motor developed at the Special Design Bureau of Electrical-Machine Engineering (Tomsk) [2].

The structural form of such a motor is shown in Fig. 1. The stator 1 of the self-braking electric motor is made as in a conventional machine. The rotor core has two parts: the braking part 4 and the main part 2, separated by a nonmagnetic spacer 3. The electromagnetic insert (armature) 5 is connected to the fan, which simultaneously serves as the brake disk and can move in the axial direction along splines on the shaft. The main part of the rotor stack is assembled from ordinary laminations, while the laminations of the braking part are stamped from steel 2–3 mm thick and have the slot configuration shown in Fig. 2. After pressing and casting the rotor with aluminum, the braking part of the rotor is turned so that the teeth are separated from one another (Fig. 2).

Fig. 1. Design of a self-braking electric motor with an electromagnetic insert

Fig. 2. Slot configuration of the braking part

When voltage is applied to the stator winding, part of the main flux \(\Phi_r\) passes through the special part of the rotor, the air gap between the electromagnetic insert and the end face of the special part, and closes through the electromagnetic insert as through a yoke. The leakage flux \(\Phi_s\) of the short-circuited ring of the special part of the rotor also passes into the electromagnetic insert through the air gap. Under the action of the electromagnetic forces that arise in this process, the insert, overcoming the force of the spring, is attracted to the end face of the special part of the rotor and pulls the fan along with it. The fan moves away from the brake shoes and the motor is released from braking.

To study the electromechanical transient processes of such motors, it is necessary to formulate equations describing the operation of the motor during starting and braking. These equations must describe both the process of switching the motor on to the network and the process of disconnecting it from the network. In addition, they must take into account that during transient processes the attractive force produced by the special (braking) part of the rotor varies in magnitude.

In writing the voltage-balance equations, it is assumed that the parameters of the stator winding \(r_1\), \(x_1\), and the active resistance of the rotor \(r_2'\) do not depend on the magnitude of the air gap \(\delta\) between the special part of the rotor and the electromagnetic insert, while the air gap affects only the inductive reactances of mutual induction \(x_m\) and leakage of the rotor winding \(x_2'\). For this purpose, in order to find the value of the attractive force \(F_{\text{pr}}\), the inductive reactances \(x_m\) and \(x_2'\) are divided into components: \(x_{m0}\) and \(x_{20}'\)—the inductive reactances of mutual induction and leakage of the main part of the rotor; \(x_{mg}\) and \(x_{2g}'\)—the inductive reactances of mutual induction and leakage of the special part of the rotor.

Taking the above into account, the parameters of the self-braking electric motor may be written in relative units in the following form:

\[ x_s = x_1 + x_{m0} + x_{mg} \]
— total inductive reactance of the stator winding;

\[ x_2 = x_{20}' + x_{2g}' + x_{m0} + x_{mg} \]
— total inductive reactance of the rotor winding;

\[ x_m = x_{m0} + x_{mg} \]
— mutual-induction reactance;

\[ r_1 \ \text{and} \ r_2' \]
— active resistances of the stator and rotor windings.

The voltage-balance equations of the rotor winding, written in a synchronously rotating coordinate system, have the form \([1, 3]\):

\[ \frac{d\Psi_{r\alpha}}{dt} = - r_2 i_{r\alpha} + \Psi_{r\beta} - \omega \Psi_{r\beta}; \]

\[ \frac{d\Psi_{r\beta}}{dt}=-r_2 i_{r\beta}-\Psi_{r\alpha}+\omega\Psi_{r\alpha}; \]

\[ \begin{aligned} \frac{di_{r\alpha}}{dt}&=-y_1 i_{r\alpha}+y_3\Psi_{r\alpha}-y_2\omega\Psi_{r\beta}+i_{r\beta}-y_4U_{s\alpha};\\ \frac{di_{r\beta}}{dt}&=-y_1 i_{r\beta}+y_3\Psi_{r\beta}+y_2\omega\Psi_{r\alpha}-i_{r\alpha}-y_4U_{s\beta};\\ i_{s\alpha}&=\frac{1}{x_m}\left(\Psi_{r\alpha}-x_2 i_{r\alpha}\right);\\ i_{s\beta}&=\frac{1}{x_m}\left(\Psi_{r\beta}-x_2 i_{r\beta}\right), \end{aligned} \tag{1} \]

where

\[ y_1=\frac{r_1x_r+x_sr_2}{x_sx_r-x_m^2}; \]

\[ y_2=\frac{x_s}{x_sx_r-x_m^2}; \]

\[ y_3=\frac{r_1}{x_sx_r-x_m^2}; \]

\[ y_4=\frac{x_m}{x_sx_r-x_m^2}; \]

The electromagnetic torque is determined by the equality

\[ M=\Psi_{r\beta}i_{r\alpha}-\Psi_{r\alpha}i_{r\beta}. \tag{2} \]

When the electric motor operates from a supply with constant voltage and frequency, the components of the voltage vector in the coordinate system under consideration are constant quantities and are equal to

\[ \begin{aligned} U_{s\alpha}&=U_m\cos\alpha_0,\\ U_{s\beta}&=U_m\sin\alpha_0, \end{aligned} \tag{3} \]

where \(\alpha_0\) is the initial phase.

When the motor is disconnected from the supply and braking takes place, the voltage-balance equations take the form:

\[ \frac{d\Psi_{r\alpha}}{dt}=-r_2 i_{r\alpha}+\Psi_{r\beta}-\omega\Psi_{r\beta}; \]

\[ \frac{d\Psi_{r\beta}}{dt}=-r_2 i_{r\beta}-\Psi_{r\alpha}+\omega\Psi_{r\alpha}; \]

\[ \begin{aligned} \frac{di_{r\alpha}}{dt}&=-y_1'i_{r\alpha}-y_2'\omega\Psi_{r\beta}+i_{r\beta};\\ \frac{di_{r\beta}}{dt}&=-y_1'i_{r\beta}+y_2'\omega\Psi_{r\alpha}-i_{r\alpha};\\ i_{r\alpha}&=\frac{\Psi_{r\alpha}}{x_r};\\ i_{r\beta}&=\frac{\Psi_{r\beta}}{x_r}, \end{aligned} \tag{4} \]

where

\[ y_1'=\frac{r_r}{x_r}, \]

\[ y_2'=\frac{1}{x_r}. \]

The electromagnetic torque when the motor is disconnected from the network is equal to zero,

\[ M=0. \tag{5} \]

The magnitude of the attractive force produced by the braking part of the rotor can be determined from the following equation:

\[ F_{\mathrm{pr}}=\frac{\Phi^2}{2\mu_0 S}, \tag{6} \]

where

\(\Phi\) is the magnetic flux in the air gap between the special part of the rotor and the electromagnetic insert;

\(S\) is the cross-sectional area of the end face of the magnetic core of the special part of the rotor to which the electromagnetic insert is attracted.

From the operating principle of the braking device it is evident that two fluxes participate in producing the attractive force: \(\Phi_s\), the leakage flux of the short-circuited ring of the special part of the rotor, and \(\Phi_t\), the part of the main flux closing through the special part of the rotor and the electromagnetic insert. Therefore the magnetic flux can be represented in the following form:

\[ \Phi=\Phi_s+\Phi_t . \tag{7} \]

In order to take into account the influence of the magnitude of the attractive force on the electromechanical transient processes of the self-braking electric motor and to establish the character of the change in this force during transient processes, we write equation (7) in terms of components along the two axes \(\alpha\) and \(\beta\). Since in relative units the flux is equal to the flux linkage, equation (7) in the \(\alpha\) and \(\beta\) axes will have the following form:

\[ \Psi_{r\alpha g} = \left( x_{rg}+x_{mg}-\frac{x_{mg}x_r}{x_m} \right)i_{r\alpha} + \frac{x_{mg}}{x_m}\Psi_{r\alpha}; \]

\[ \Psi_{r\beta g} = \left( x_{rg}+x_{mg}-\frac{x_{mg}x_r}{x_m} \right)i_{r\beta} + \frac{x_{mg}}{x_m}\Psi_{r\beta}, \tag{8} \]

where

\[ \left( x_{rg}+x_{mg}-\frac{x_{mg}x_r}{x_m} \right)i_{r\alpha} \quad \text{and} \quad \left( x_{rg}+x_{mg}-\frac{x_{mg}x_r}{x_m} \right)i_{r\beta} \]

are the flux-linkage components along the \(\alpha\) and \(\beta\) axes corresponding to the leakage flux of the short-circuited ring of the special part of the rotor \(\Phi_s\);

\[ \frac{x_{mg}}{x_m}\Psi_{r\alpha} \quad \text{and} \quad \frac{x_{mg}}{x_m}\Psi_{r\beta} \]

are the flux-linkage components along the \(\alpha\) and \(\beta\) axes corresponding to the part of the main flux \(\Phi_t\).

Knowing the flux linkage of the special part of the rotor, we write equation (6) in the following form:

\[ F_{\mathrm{pr}\alpha}=\frac{\Psi_{r\alpha g}^{2}}{2\mu_0 S}, \]

\[ F_{\mathrm{pr}}=\frac{\Psi^{2} r_{\delta} g}{2\mu_{0}S}. \tag{9} \]

In studying electromechanical transient processes, along with the voltage-balance equations of the electrical circuits, it is necessary to have the equation of motion of the rotor. Let us consider the process of starting and braking the self-braking electric motor when the electromagnetic insert is in two extreme positions, depending on which the equations of motion of the rotor have one form or another.

Starting of the self-braking motor. The starting of the self-braking electric motor can be divided into the following two stages: the instant of breakaway and the instant of acceleration of the motor.

At the instant when the motor starts from rest, the magnitude of the attraction force \(F_{\mathrm{pr}}\) is less than the initial compression force \(N_{0}\); therefore the braking torque produced by the spring is not equal to zero,

\[ M_{\mathrm{T}}=N_{\mu}R, \tag{10} \]

\[ N=N_{0}-F_{\mathrm{pr}}, \]

where \(\mu\) is the coefficient of friction.

The equation of motion in this case has the form

\[ M=M_{\mathrm{T}}+M_{\mathrm{c}}+I\frac{d\omega}{dt}. \tag{11} \]

Here \(M_{\mathrm{c}}\) is the load-resistance torque.

At the instant of acceleration of the motor, the attraction force \(F_{\mathrm{pr}}>N_{0}+C\delta_{x}\), and the electromagnetic insert is attracted to the end face of the special part of the rotor. The inequality

\[ F_{\mathrm{pr}}>N_{0}+C\delta_{x} \tag{12} \]

is the condition for releasing the brake.

\(C\) is the stiffness coefficient of the spring,

\(\delta_{x}\) is the amount of compression of the spring under the action of the attraction force when the insert is attracted.

The equation of motion of the rotor during acceleration of the self-braking motor can be written in the form

\[ M=M_{\mathrm{c}}+I\frac{d\omega}{dt}. \tag{13} \]

Braking of the self-braking electric motor

Braking of the self-braking motor is carried out when it is disconnected from the supply network. In this case the following stages of braking can be distinguished: the instant of disconnection of the motor from the network, the instant of mechanical braking, and the instant of stopping.

At the instant of disconnection of the motor from the network, the electromagnetic insert is attracted to the end face of the rotor, since the attraction force \(F_{\mathrm{pr}}\) is greater than the spring compression force \(N_{0}+C\delta_{x}\), and the braking torque is equal to zero. This is explained by the fact that the magnitude of the attraction force \(F_{\mathrm{pr}}\) is determined by the fluxes \(\Phi_{s}\) and \(\Phi_{\mathrm{T}}\), which, when the electric motor is disconnected from the network, cannot instantaneously change their magnitudes to zero. Therefore, as long as the attraction force \(F_{\mathrm{pr}}>N_{0}+C\delta_{x}\) exceeds the spring compression force, the electromagnetic insert remains in the attracted position. The equation of motion of the rotor for this instant has the form

\[ 0=M_{\mathrm{c}}+I\frac{d\omega}{dt}. \tag{14} \]

After the attraction force \(F_{\mathrm{pr}}\) becomes less than the spring compression force \(N_{0}+C\delta_{x}\), the electromagnetic insert, because of the small magnitude

the stroke almost instantaneously passes from the position in which it is attracted to the end face of the rotor to the position in which it is pressed back by the spring.

After this, the instant of mechanical braking begins. At the instant of mechanical braking, the magnitude of the braking torque is equal to

\[ M_{\mathrm{t}}=N\mu R, \tag{15} \]

where

\[ N=N_0-F_{\mathrm{pr}}, \quad \text{and } F_{\mathrm{pr}}<N_0 . \]

The equation of motion for this stage has the form

\[ 0=M_{\mathrm{t}}+M_{\mathrm{s}}+i\frac{d\omega}{dt}. \tag{16} \]

When the electric motor stops, the braking torque is equal to

\[ M_{\mathrm{t}}=N_0\mu R . \tag{17} \]

The attractive force \(F_{\mathrm{pr}}\) is equal to zero.
The equation of motion takes the form

\[ 0=M_{\mathrm{t}}+M_{\mathrm{s}} . \tag{18} \]

The set of voltage-balance equations and rotor-motion equations corresponding to the various positions of the electromagnetic insert during starting and braking of the electric motor makes it possible to investigate the electromechanical transient processes. The complete system of equations is nonlinear, and therefore it is most expedient to carry out the investigation by modeling on an analog computer. The block diagram of the model is presented in Fig. 3.

Fig. 3. Block diagram of the model

Modeling of the voltage-balance equations in the electric motor is carried out by means of amplifiers Л1–Л10 and nonlinear blocks Н1–Н4. Amplifiers Л11–Л15 and nonlinear blocks Н15–Н16 solve equations associated with the magnitude of the attractive force and make it possible to carry out

effect the transition from one equation of rotor motion to another. Operational relays OPR1-OPR5 make it possible to carry out the transition from the system of voltage-equilibrium equations during starting to the system of equations during braking.

When the electric motor is started, the operational relays OPR1-OPR5 are de-energized. In this case equations 1, 2, 7, 11-13 are solved. The transition from equation (11) to equation (13) is carried out by means of a comparing device containing amplifier L15 and relay R1.

When the electric motor is braked, voltage is applied to the operational relays OPR1-OPR5. In this case the contacts of relays R3, R4 open, R1, R2 switch over, and R5 close. With this position of the contacts, equations 4, 7, 14-16 are solved. Oscillograms of starting and braking of the electric motor at different braking torques are given in Fig. 4.

Legend to Fig. 4:

• solid line: for \(M_t/M_n = 0.3\)
• dashed line: for \(M_t/M_n = 0.42\)
• dash-dot line: for \(M_t/M_n = 1.18\)

Fig. 4. Oscillograms of starting and braking at different values of braking torque

From the oscillogram of starting and braking it is seen that, as the braking torque increases, the stopping time \(t_{\mathrm{ost}}\) decreases. The braking torque has no effect on the acceleration time \(t_p\), which can be explained by the very rapid (almost instantaneous) operation of the braking device when the electric motor is started. The stopping time \(t_{\mathrm{ost}}\) of the electric motor is the sum of two components: \(t_{\mathrm{sr}}\), the operating time of the braking device when the electric motor is disconnected from the supply, and \(t_t\), the time of mechanical braking.

A comparison of calculated and experimental data for electric motors AO2-21-29, 4A112-49, and 4A132-29 showed that the error in calculating tractive forces, starting time, and stopping time does not exceed 15-20%.

The proposed mathematical model of the self-braking electric motor can be used in studying transient processes and in designing self-braking electric motors with an electromagnetic insert.

References

1. A. D. Hildebrand, M. G. Zhuk, V. M. Kirpichnikov. Solving Engineering Problems on Electronic Computers. UPI Publishing House, Sverdlovsk, 1970.

2. E. M. Guselnikov, B. S. Tsukerman. Self-Braking Electric Motors. Energiya, 1971.

3. M. M. Sokolov, L. P. Petrov, L. B. Masandilov, B. A. Ladenzon. Electromagnetic Transient Processes in an Asynchronous Electric Drive. Energiya, 1967.