CALCULATION OF THE TRACTION FORCES OF SELF-BRAKING ASYNCHRONOUS MOTORS WITH ELECTROMAGNETIC INSERTS

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)

The main parameter of self-braking electric motors (SBEM) is the traction force developed by the braking part of the rotor. It determines not only the magnitude of the braking torque and the starting and stopping time of the electric motor, but also the weight and energy characteristics of the SBEM.

Therefore, in designing an SBEM it is necessary to know how the traction force depends on the geometrical dimensions of the braking part of the rotor and what the nature is of the change in the force during starting and braking of the electric motor.

Taking into account the design features of the braking part of the rotor of the electric motor [1], the magnitude of the traction force can be determined from Maxwell’s formula

\[ F_{\mathrm{tr}} = \frac{\Phi^2}{2\mu_0 S}, \tag{1} \]

where

\[ \begin{aligned} \Phi &\text{ is the magnetic flux in the air gap between the braking part of the rotor and the electromagnetic insert;}\\ \mu_0 &\text{ is the magnetic permeability of air;}\\ S &\text{ is the cross section of the magnetic circuit of the braking part of the rotor.} \end{aligned} \]

From the operating principle of the braking device of an SBEM with an electromagnetic insert [1, 5], it is known that two magnetic fluxes participate in producing the traction force, namely: \(\Phi_s\), the leakage flux of the short-circuited ring of the braking part of the rotor, and \(\Phi_T\), a part of the main magnetic flux of the stator. Therefore the magnetic flux \(\Phi\) in the air gap between the braking part of the rotor and the electromagnetic insert can be represented in the form of two components:

\[ \Phi = \Phi_s + \Phi_T . \tag{2} \]

The magnitude of the flux \(\Phi_s\) depends on the current flowing in the short-circuited ring and on the reluctance of the magnetic circuit of the braking part of the electric motor. The magnitude of the flux \(\Phi_T\) is determined by the geometrical dimensions and the reluctance of the magnetic circuit of the braking part of the electric motor.

At the moment of starting the SBEM, when the rotor is still stationary, a large current flows in the short-circuited ring of the braking part of the rotor, producing the magnetic flux \(\Phi_s\), which ensures the attraction of the electromagnetic insert to the end face of the rotor. As the electric motor accelerates, the current in the short-circuited ring and the magnetic flux \(\Phi_s\) decrease.

At the same time, due to the reduction of the reverse effect of the rotor windings on the stator windings, the component of the main flux \(\Phi_T\) increases, which ensures that the electromagnetic insert is held in the position attracted to the end face of the rotor [1]. Taking the foregoing into account, it may be concluded that the nature of the change in the tractive force during the starting process is substantially influenced by the leakage flux of the short-circuited ring \(\Phi_S\), while the magnitude of the tractive force in the steady operating mode of the electric motor is influenced by the flux \(\Phi_T\). In order to determine the nature of the change in tractive force during the starting process of the SED, it is necessary to set up a system of equations describing the transient electromagnetic processes. The most convenient method for investigating transient processes in asynchronous motors is the method of mathematical modeling. In order for the system of equations to be suitable for mathematical modeling, it is necessary to write the equations in the coordinate axes \(\alpha\) and \(\beta\), which in the general case may rotate at an arbitrary speed. Various forms of writing systems of equations are known [2], for which the transition from one form to another is carried out with the aid of equations for the stator and rotor flux linkages, written in the coordinate axes \(\alpha\) and \(\beta\):

\[ \begin{aligned} \omega_0 \Psi_{s\alpha} &= x_s i_{s\alpha} + x_m i_{r\alpha},\\ \omega_0 \Psi_{s\beta} &= x_s i_{s\beta} + x_m i_{r\beta},\\ \omega_0 \Psi_{r\alpha} &= x_r i_{r\alpha} + x_m i_{s\alpha},\\ \omega_0 \Psi_{r\beta} &= x_r i_{r\beta} + x_m i_{s\beta}, \end{aligned} \tag{3} \]

where

\(\Psi_{s\alpha}\) and \(\Psi_{s\beta}\); \(\Psi_{r\alpha}\) and \(\Psi_{r\beta}\)—are the flux linkages of the stator and rotor in the axes \(\alpha\) and \(\beta\).

\[ x_s = x_1 + x_m \]
—the total inductive reactance of the stator winding in relative units;

\[ x_r = x'_2 + x_m \]
—the total inductive reactance of the rotor winding in relative units;

\(r_1\) and \(r_2\)—active resistances of the stator and rotor windings in relative units;

\(\omega_0\)—synchronous rotational speed (angular frequency).

Let us write the system of equations (3), taking into account the assumptions adopted in [5]:

\[ \begin{aligned} \omega_0 \Psi_{s\alpha} &= (x_1 + x_{m0} + x_{mg}) i_{s\alpha} + (x_{m0} + x_{mg}) i_{r\alpha},\\ \omega_0 \Psi_{s\beta} &= (x_1 + x_{m0} + x_{mg}) i_{s\beta} + (x_{m0} + x_{mg}) i_{r\beta},\\ \omega_0 \Psi_{r\alpha} &= (x'_2 + x'_{2g} + x_{m0} + x_{mg}) i_{r\alpha} + (x_{m0} + x_{mg}) i_{s\alpha},\\ \omega_0 \Psi_{r\beta} &= (x'_2 + x'_{2g} + x_{m0} + x_{mg}) i_{r\beta} + (x_{m0} + x_{mg}) i_{s\beta}, \end{aligned} \tag{4} \]

where

\(x_{m0}\) and \(x'_{20}\)—inductive reactances of mutual induction and leakage of the main part of the rotor;

\(x_{mg}\) and \(x'_{2g}\)—inductive reactances of mutual induction and leakage of the braking part of the rotor.

Carrying out transformations of the system of equations, we obtain the flux linkages falling on the braking part of the rotor, which correspond to the fluxes \(\Phi_S\) and \(\Phi_T\):

\[ \Psi_{r\alpha g} = \left( x_{mg} + x'_{2g} - \frac{x_r x_{mg}}{x_m} \right)i_{r\alpha} + \frac{x_{mg}}{x_m}\Psi_{r\alpha}, \]

\[ \Psi_{r\beta g} = \left( x_{mg} + x'_{2g} - \frac{x_r x_{mg}}{x_m} \right)i_{r\beta} + \frac{x_{mg}}{x_m}\Psi_{r\beta}. \tag{5} \]

Since in relative units the flux is equal to the flux linkage, the components of the fluxes \(\Phi_T\) and \(\Phi_S\) along the axes \(\alpha\) and \(\beta\) are written in the form

\[ \begin{aligned} \Phi_{r\alpha} &= -\frac{x_{m\sigma}}{x_m}\Psi_{r\alpha};\\ \Phi_{r\beta} &= \frac{x_{m\sigma}}{x_m}\Psi_{r\beta};\\ \Phi_{s\alpha} &= \left(x_{mg}+x'_{2g}+\frac{x_r x_{m\sigma}}{x_m}\right)i_{r\alpha};\\ \Phi_{s\beta} &= \left(x_{mg}+x'_{2g}-\frac{x_r x_{mg}}{x_m}\right)i_{r\beta}. \end{aligned} \tag{6} \]

Using equations (1), (2), and (6), we obtain an expression for the tractive force

\[ F_{\mathrm{tr}}= \frac{ \left[\left(x_{mg}+x'_{2g}-\frac{x_r x_{mg}}{x_m}\right)i_{r\alpha} +\frac{x_{mg}}{x_m}\Psi_{r\alpha}\right]^2 + \left[\left(x_{mg}+x'_{2g}-\frac{x_r x_{mg}}{x_m}\right)i_{r\beta} +\frac{x_{mg}}{x_m}\Psi_{r\beta}\right]^2 } {2\mu_0 S}. \tag{7} \]

The structural diagram of the model, which makes it possible to simulate equation (7), is presented in Fig. 1.

Fig. 1. Structural diagram of the model of the braking part of the rotor

Oscillograms of the tractive force at the moment of starting of the self-braking electric motor AO2-21-2E with an electromagnetic insert are given in Fig. 2. Here \(F_m\) is the impact value of the tractive force during starting, and \(F_{\mathrm{ud}}\) is the magnitude of the holding force of the electromagnetic insert in the established operating mode of the electric motor.

Fig. 2. Oscillogram of the tractive force of the braking part of the rotor at the moment of starting of the electric motor AO2-21-2E

Let us consider, for the AO2-21-23 electric motor, how the forces \(F_m\) and \(F_{\mathrm{ud}}\) change as a function of the length of the braking part of the rotor. From expressions (1, 7) it is evident that the tractive force is determined by the cross section of the magnetic circuit of the braking part of the rotor and by the magnetic flux, whose magnitude depends on the inductive reactances of mutual induction \(x_{mg}\) and \(x_m\), and of leakage \(x'_{2g}\) and \(x_r\).

The change in the ratio \(\dfrac{x_{mg}}{x_m}\) is determined by the change in the length of the braking part of the rotor; therefore the expressions for the inductive reactances of the main and braking parts of the rotor, with zero air gap between the end face of the rotor and the electromagnetic insert, can be written in the following form [4]:

\[ x_{m0}=\frac{4mf_1}{\pi}\cdot\frac{\tau \cdot w_1^2 k_{ob1}^2}{p}\cdot \lambda_{\delta} l_{10}; \tag{8} \]

and

\[ x_{mg}=\frac{4mf_1}{\pi}\cdot\frac{\tau \cdot w_1^2 k_{ob1}^2}{p}\lambda_{\delta g(0)} l_{1g}, \tag{9} \]

\[ \frac{x_{mg}}{x_m} = \frac{x_{mg}}{x_{m0}+x_{mg}} = \frac{\lambda_{\delta g(0)} l_{1g}} {\lambda_{\delta} l_{10}+\lambda_{\delta g0} l_{1g}}, \tag{10} \]

where

\[ \begin{aligned} w_1&\text{ is the number of conductors per phase of the stator winding;}\\ k_{ob1}&\text{ is the winding factor of the stator winding;}\\ f_1&\text{ is the supply-network frequency;}\\ \tau&\text{ is the pole pitch;}\\ p&\text{ is the number of pole pairs;}\\ l_{10},\, l_{1g}&\text{ are the stator-stack lengths corresponding to the main and braking parts;}\\ \lambda_{\delta}=\frac{\mu_0}{k_{\mu} k_{\delta}\delta} &\text{ is the magnetic permeance in the air gap between the stator and rotor;}\\ k_{\mu}&\text{ is the saturation coefficient;}\\ k_{\delta}&\text{ is Carter’s coefficient for the main part;}\\ \delta&\text{ is the air gap between the rotor and stator;}\\ \lambda_{\delta g(0)}=\frac{\mu_0}{k_{\mu} k_{\delta g}\delta} &\text{ is the magnetic permeance in the air gap between the stator and the braking part of the rotor at zero air gap between the braking part of the rotor and the electromagnetic insert.} \end{aligned} \]

For the AO2-21-23 electric motor under investigation, with zero air gap between the braking part of the rotor and the electromagnetic insert,

\[ \lambda_{\delta g(0)} \simeq \lambda_{\delta}. \tag{11} \]

Therefore expression (10) can be written as

\[ \frac{x_{mg}}{x_m} = \frac{l_{1g}}{l_{10}+l_{1g}} = \frac{l_{1g}}{l_1}, \tag{12} \]

where \(l_1\) is the total length of the stator stack.

Figure 3 presents the dependences of the tractive forces \(F_m\) and \(F_{\mathrm{ud}}\) on the length of the braking part of the rotor. From Fig. 3 it is evident that, as the length of the braking part of the rotor changes, the holding force \(F_{\mathrm{ud}}\) changes substantially, whereas the impact value of the attraction force \(F_m\) changes only slightly. That is, at the moment of starting of the electric motor, owing to the leakage fluxes of the short-circuited rotor ring \(\Phi_s\), the electromagnetic insert will be attra-

Fig. 3. Dependence of the traction forces on the length of the braking part of the rotor

...to develop even if the length of the braking part of the rotor is small. In the steady-state mode, however, the magnitude of the flux \(\Phi_\delta\) is small, and the holding force of the insert \(F_{\mathrm{hold}}\) is determined mainly by the flux \(\Phi_t\), whose magnitude depends on the length of the braking part of the rotor.

The proposed method for calculating the traction force makes it possible to establish the nature of the variation of the traction force during the starting process of the electric motor and to determine the optimal dimensions of the braking part of the rotor of self-braking electric motors.

References

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

2. A. D. Hildebrand, M. G. Zhuk, V. M. Kirpichnikov. Solving Engineering Problems on Electronic Computers. Part I, UPI, 1970.

3. M. M. Sokolov, L. P. Petrov, L. B. Masandilov, V. A. Ladenzon. Electromagnetic Transient Processes in Electric Drives. Energiya, 1967.

4. P. S. Sergeev, N. V. Vinogradov, F. A. Goryainov. Design of Electrical Machines. Energiya, 1969.

5. L. K. Burulko, N. G. Guselnikov, E. V. Kononenko, G. V. Chetverta. Study of electromechanical transient processes in a self-braking electric motor with an electromagnetic insert. Proceedings of TPI. Present collection.