Some Features of Capacitor Self-Excitation of Single-Phase Induction Machines

G. A. SIPAILOV, Yu. A. ROMANOV, Yu. I. PEREZHIROV

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

At present, the phenomenon of electromagnetic instability together with capacitor self-excitation is used in electric drives for capacitor braking of induction motors. Moreover, the more rational braking method is considered to be a combined one, combining both capacitor and dynamic braking. In capacitor braking, the kinetic energy of the rotating rotor is converted into electromagnetic energy, released in the form of heat mainly in the steel and in the stator winding.

Capacitor braking is most effective when the capacitance is connected asymmetrically to the phases of the motor stator. In this case the nonuniform distribution of currents causes different heating of the stator phases and leads to the appearance of additional losses in the rotor due to the occurrence of a counter-rotating magnetic field.

The simplest version of the circuit for asymmetric capacitor self-excitation (or braking) is shown in Fig. 1, where the capacitance is connected to two stator phases, while the third phase is free. An approximate analysis of the boundaries of the self-excitation region of an induction machine in two phases showed that the capacitance required for excitation of a single-phase machine is determined from the inequality [1]:

Fig. 1. Diagram of the electric circuit of a single-phase induction machine

\[ C_1 > \frac{2}{\omega\left(x_{d1}+x'_{d1}\right)}, \tag{1} \]

where

\[ \omega \]

is the angular frequency;

\[ x_{d1} \]

is the inductive reactance of the stator of a single-phase machine;

\[ x'_{d1} \]

is the transient reactance of the stator of a single-phase machine.

For the convenience of comparing the selection of the required capacitance for three- and single-phase machines, it is expedient to consider the operation of a single-phase machine on two phases of a three-phase one, between the parameters of which the known relations exist [2]:

\[ x_{d1}=2x_d,\qquad x'_{d1}=2x'_d . \tag{2} \]

Taking (2) into account, expression (1) is transformed into

\[ C_1>\frac{1}{\omega\left(x_d+x'_d\right)} . \tag{3} \]

The total capacitance of the capacitor banks in a delta circuit, required for the excitation of a three-phase machine, can be determined from the relation:

\[ C_{\Delta}>\frac{1}{\omega x_d}. \tag{4} \]

Comparing expressions (3) and (4), we see that

\[ \frac{C_{\Delta}}{C_1}=\left(1+\frac{x'_d}{x_d}\right). \tag{5} \]

As follows from (5), the required capacitance of the capacitor banks for the excitation of three-phase and single-phase induction machines is approximately the same if \(x'_d \ll x_d\). Under capacitor braking, when significant saturation of the steel takes place, the expression \(x'_d/x_d\) has a substantial effect. As experimental studies have shown, with increasing \(C_{\Delta}\) and \(C_1\), the saturation of the steel increases; in this case the value \(C_{\Delta}/C_1\) changes and reaches 1.3. With single-phase capacitor self-excitation, the counter-rotating magnetic field increases energy losses; therefore it is of interest to study the physical processes caused by the electromagnetic instability of single-phase induction machines.

The study of capacitor self-excitation in time is a difficult problem. In particular, when solving differential equations it becomes necessary to take the saturation of the steel into account. The combination of mathematical modeling with experimental work on a mock-up specimen of an induction generator facilitates the task of the investigation.

Capacitor self-excitation of an induction machine is accomplished by connecting the stator winding to a capacitor bank. Let us analyze the process of self-excitation of a single-phase induction generator on the mathematical model developed by the authors [3]. In the block diagram of the model, self-excitation takes place, with the corresponding parameters, after the “start” command is given. The appearance of an initial current in the stator circuit of the physical model may be a consequence of residual magnetic flux, a random charge on the capacitance, or external electromagnetic actions. On the experimental installation it was not possible to obtain a voltage due to residual flux of less than 0.5% of the nominal value; in this case self-excitation occurred over a longer time compared with the variant in which the self-excitation cycles follow one after another. The presence of residual flux in the machine or an initial voltage on the capacitance shortens the excitation time of the generator. Self-excitation in the block diagram of the model occurs at zero initial values of the voltage on the integrators, that is, from external electromagnetic disturbances.

With the appearance of initial energy in the circuit, a continuous exchange takes place between the machine and the electric field of the capacitance. Gradually this energy is dissipated in losses. For certain ratios of the parameters of the oscillatory circuit (when the real part of the complex roots of the characteristic equation is greater than zero), energy is drawn from the rotor. With linear parameters of the machine–capacitance system and a constant rotor speed, the increase in current and voltage theoretically could continue to infinity. In reality, during self-excitation the growth of current is limited by the nonlinearity of the circuit (in an induction machine, by saturation of the steel). Obviously, the appearance of a steady-state regime during self-excitation occurs when all the energy corresponding to the mechanical torque on the rotor shaft is dissipated in the active resistances of the machine circuits.

With any change in the current of the stator circuit of a single-phase induction generator, a transformer emf is induced along the longitudinal axis of the rotor winding; its current impedes the penetration of the magnetic flux into the rotor circuit and displaces it outside this circuit. The increase in the amplitude of the transverse component of the rotor current is clearly delayed (Fig. 2). This

Fig. 2. Change in the free components of the rotor current \(i_{ar}\), \(i_{\beta r}\) during self-excitation of a single-phase induction machine

can be explained by the delayed penetration of the armature-reaction flux into the rotor winding circuits. An increase in the stator current is accompanied by damping of the flux by the rotor winding. Such a process would continue indefinitely if \(R_r = 0\). The flux would never penetrate into the rotor, and in the machine there would exist a pulsating flux from random currents, there would be no asynchronous torque, and consequently no self-excitation. But in real machines \(R_r > 0\). This leads to the gradual penetration of the stator-reaction flux into the rotor-winding circuits.

The increase in the resultant flux vector in the gap is not monotonic (Fig. 3); clearly expressed pulsations of double frequency are present. Moreover, the amplitude of these pulsations during the excitation process first increases, reaches its maximum in the period of the greatest rate of growth of the generator voltage, and decreases in the established regime. The presence of flux pulsations in time can be explained by the ellipsoidal character of rotation of the field in the machine air gap. The change in flux in the air gap during the excitation process is clearly illustrated by the hodograph of the vector \(\psi_\delta\) in Fig. 4. The motion of the geometric end of the vector \(\psi_\delta\) occurs along the trajectory of an ellipse, with the major axis occupying an intermediate position between the longitudinal and transverse axes of the machine \(\alpha\) and \(\beta\). This indicates that, during the excitation process, the maximum flux in the machine air gap does not correspond-

Fig. 3. Oscillogram of the voltage across the capacitor and of the flux in the air gap of a single-phase induction machine

Fig. 4. Oscillogram of the flux vector

is, in time, the maximum flux in the stator. Only in the excited state does the hodograph of the flux vector approach a circle, and the pulsations at double frequency weaken.

Part of the energy of the machine’s magnetic flux participates in charging the capacitor. Obviously, it may be assumed that the amplitude of the pulsations of the resulting flux vector in the air gap is proportional to the maximum energy stored in the capacitance at the instant \(u_c = U_{cm}\) (Fig. 5). From the oscillogram in Fig. 5 it is evident that, as the stator current decreases, ...

Fig. 5. Calculated curves of current, voltage, and flux under established self-excitation of an A. M.

the transverse component of the rotor current $i_{\beta r}$, tending to maintain the flux constant; however, it changes and, at the moment $i_\alpha = 0$, reaches its minimum. As the stator current increases from 0 to $I_m$, the flux $\psi_\delta$ changes from $\psi_{\min}$ to $\psi_{\max}$. Calculations show that flux pulsations are also observed with a superconducting short-circuited rotor winding. As was to be expected, the component $i_{\alpha r}$ is in phase opposition to the stator current. Only the presence of flux in the rotor determines the existence of the transverse component $i_{\beta r}$, which tends to maintain the flux constant.

Thus, the investigations carried out have shown that one of the most rational methods of capacitor braking of asynchronous machines is single-phase braking (on two phases of a three-phase machine). In this case, when the steel is significantly saturated, a smaller capacitance is required than with three-phase capacitor braking; moreover, the braking effect is increased owing to the increased energy losses from the counter-rotating magnetic field. The counter-rotating magnetic field is manifested to the greatest extent in the process of self-excitation, when the amplitude of the stator current and voltage increases. The amount of energy participating in the exchange between the machine and the capacitance is much less than the energy stored in the magnetic field of the motor.

REFERENCES

1. Yu. A. Romanov, G. A. Sipailov. On self-excitation of a single-phase asynchronous impulse generator. Electron Accelerators. Proceedings of the VII Interuniversity Conference on Electron Accelerators, Issue 3, 1970.

2. K. P. Kováč, I. Rats. Transient Processes in AC Machines. Gosenergoizdat, 1963.

3. Yu. A. Romanov, G. A. Sipailov. Analysis of the operation of an asynchronous impulse generator using an analog computer. Izv. TPI, vol. 212, 1971.