SELF-EXCITATION OF INDUCTION MOTORS FED FROM A STATIC FREQUENCY TRIPLER

Ya. V. PETROV, A. I. ZAITSEV

(Presented by the scientific seminar of the Electromechanical Faculty)

A source of 150 cycles/sec current for three-phase low-power induction motors, for example, a hand-held electric tool, may be a static electromagnetic frequency tripler with series capacitive compensation at the output. However, when the motor is started, the latter often becomes self-excited and, operating simultaneously as a motor and as a generator of reduced-frequency currents, does not reach its normal speed. The resulting torque on the shaft decreases (practically to zero for an unloaded motor) and oscillates about a certain mean value as a consequence of the rotor rocking relative to the corresponding mean speed (Fig. 1)¹.

Fig. 1.

Currents of the fundamental and reduced frequencies flow through the elements of the installation, which causes a sharp distortion of the voltage and current waveforms of the beating type, intense heating, and overvoltages of the elements.

¹ The mechanical characteristics of the motor were obtained by the method developed by the Department of Electrification of Industrial Enterprises of the Tomsk Polytechnic Institute [6].

According to our experimental data, the speed of jamming of a motor started at no load is about 40–60% of synchronous speed (the higher values refer to smaller capacitor capacitances); the voltage peaks at the motor exceed the amplitude of the tripler no-load output voltage by a factor of 1.9–2.0; the current peaks are approximately equal to the largest amplitudes of the starting current; the beat frequency is 17–18 periods/sec.

The phenomena of self-excitation of industrial-frequency induction motors fed from transformers or synchronous generators of comparable power through series-connected capacitors, and of motors in the system under consideration, are fundamentally of the same type and are distinguished by considerable complexity [1, 2, 3, 4, etc.]. The nonlinearity of the characteristics and the inevitable partial asymmetry and nonsinusoidality of the tripler output quantities increase the difficulties.

The purpose of the present work is to select the most preferable methods of eliminating self-excitation of motors connected to the tripler through series capacitors. The parameters of the tripler and of the induction motor are taken to be linear; the voltages and currents at the tripler output form three-phase symmetrical and sinusoidal systems; the transient electromagnetic process is not taken into account; the generator mode of the motor is considered independently of the motor mode. The investigation is based on the method of B. G. Zagorskii [2].

Figure 2 shows the equivalent circuit of the installation for free currents (generator mode).

Fig. 2.

The equations of the electrical equilibrium of the system according to Kirchhoff are

\[ i_1+i_m-i'_2=0; \tag{1} \]

\[ L_3\frac{di_1}{dt}-r_3 i_1+\frac{1}{C}\int i_1\,dt+r_1 i_1-L_1\frac{di_1}{dt}-L_m\frac{di_m}{dt}+r_m i_m=0; \tag{2} \]

\[ L'_2\frac{di'_2}{dt}+\frac{r'_2}{s_0}i'_2+r_m i_m+L_m\frac{di_m}{dt}=0, \tag{3} \]

where \(s_0\) is the slip of the rotor relative to the stator free-current field,

\[ s_0=\frac{\omega_0-\omega_r}{\omega_0}, \tag{4} \]

here \(\omega_0\) is the angular frequency of the stator free currents,
\(\omega_r\) is the angular speed of the rotor.

Applying algebraization of the differential equations, we rewrite (1), (2), (3):

\[ i_1+i_m-i_2'=0; \]

\[ \left(L_3p+r_3+\frac{1}{Cp}+r_1+L_1p\right)i_1+(L_mp+r_m)i_m=0; \]

\[ (L_mp+r_m)i_m+\left(L_2'p+\frac{r_2'}{s_0}\right)i_2'=0. \]

The characteristic equation of the system is obtained by setting the determinant of the system equal to zero,

\[ \Delta= \left| \begin{array}{ccc} -1 & 1 & -1\\[2mm] L_3p+r_3+\dfrac{1}{Cp}+r_1+L_1p & L_mp+r_m & 0\\[2mm] 0 & L_mp+r_m & L_2'p+\dfrac{r_2'}{s_0} \end{array} \right|=0. \tag{5} \]

After simplification,

\[ \begin{aligned} &p^3\left[L_{13}(L_2'+L_m)+L_2'L_m\right] +p^2\left[\frac{r_2'}{s_0}(L_{13}+L_m)+r_{13}(L_2'+L_m)\right.\\ &\left.\qquad\qquad\qquad\qquad\quad +r_m(L_{13}+L_2')\right] +p\left[\frac{r_2'}{s_0}(r_{13}+r_m)+\frac{1}{C}(L_2'+L_m)+r_{13}r_m\right]\\ &\qquad\qquad\qquad +\frac{1}{C}\left(\frac{r_2'}{s_0}+r_m\right)=0. \end{aligned} \tag{6} \]

For the test motor \(r_m\ll x_m\); therefore the former may be neglected, and the characteristic equation is simplified:

\[ \begin{aligned} &p^3\left[L_{13}(L_2'+L_m)+L_2'L_m\right] +p^2\left[\frac{r_2'}{s_0}(L_{13}+L_m)+r_{13}(L_2'+L_m)\right]\\ &\qquad +p\left[\frac{r_2'}{s_0}r_{13}+\frac{1}{C}(L_2'+L_m)\right] +\frac{r_2'}{s_0}\cdot\frac{1}{C}=0. \end{aligned} \tag{6'} \]

In equations (6), (6′),

\[ L_{13}=L_1+L_3,\qquad r_{13}=r_1+r_3. \]

From (6′) we determine the slip \(s_0\):

\[ s_0 = -\frac{ p^2r_2'(L_{13}+L_m)+pr_2'r_{13}+\dfrac{r_2'}{C} }{ (L_2'+L_m)\left(p^3L_\sigma+p^2r_{13}+\dfrac{p}{C}\right) }. \tag{7} \]

where

\[ L_\sigma=L_{13}+\frac{L_2'L_m}{L_2'+L_m}. \]

We separate the regions of stable operation (motor mode) and unstable operation (generator mode) of the motor by the method of \(D\)-partition with respect to one parameter. As the real part of the complex-

of the variable \(\beta\), in whose plane the indicated regions are separated, we take the variable parameter \(s_0\); the imaginary part of the variable \(\beta\) is the auxiliary parameter \(\alpha\), representing the value of the imaginary part of (7) when \(p=j\omega_0\).

In (7) we substitute \(p=j\omega_0\), separate the real part from the imaginary part, and replace the inductances and capacitance by the corresponding resistances at a frequency of 150 cycles/sec. After transformations we obtain the following equation for the boundary of \(D\)-partition with respect to the parameter \(s_0\):

\[ \begin{aligned} \beta = s_0 + j\alpha &= -\,\frac{r'_2 r_{13}} {\left(1+\dfrac{x'_2}{x_m}\right)^2 \left[r_{13}^2+\left(\omega_0^* x_e-\dfrac{x_c}{\omega_0^*}\right)^2\right]} \\ &\quad + j\, \frac{ (\omega_0^*)^2 r'_2 x_e (x_{13}+x_m) - r'_2\left[ x_c(x_{13}+x_m+x_e)-r_{13}^2-\left(\dfrac{x_c}{\omega_0^*}\right)^2 \right] } { \omega_0^*(x'_2+x_m) \left[ r_{13}^2+\left(\omega_0^* x_e-\dfrac{x_c}{\omega_0^*}\right)^2 \right] }, \end{aligned} \tag{8} \]

where \(\omega_0^*\) is the angular frequency of free oscillations in fractions of the angular frequency of the current at the output of the tripler.

Substituting the numerical values of the system parameters and taking \(\omega_0^* = 0 \div 1.0\), we plot the boundary curves of \(D\)-partition with respect to the parameter \(s_0\) for various degrees of compensation (Fig. 3; \(218\,\mu F\) is the capacitance of normal compensation, \(293\,\mu F\) is the capacitance of overcompensation). The hatched regions correspond to stable operation of the motor; the regions bounded by the curves are zones of unstable operation of the motor—zones of self-excitation.

Each curve intersects the axis of real quantities at two points, which characterize the mode at the boundary of stable operation. The values of the lower and upper limiting slips, \(s_{0\mathrm{H}}\) and \(s_{0\mathrm{B}}\), can be obtained by direct reading from Fig. 3 or determined analytically according to (8); in the latter case we have

\[ s_{0\mathrm{H},\mathrm{B}} = -\, \frac{r'_2 r_{13}} {\left(1+\dfrac{x'_2}{x_m}\right)^2 \left[ r_{13}^2+\left(\omega_{0\mathrm{H},\mathrm{B}}^* x_e-\dfrac{x_c}{\omega_{0\mathrm{H},\mathrm{B}}^*}\right)^2 \right]}, \tag{9} \]

where the limiting frequencies of self-excitation \(\omega_{0\mathrm{H}}^*\) and \(\omega_{0\mathrm{B}}^*\) are found from the expression for the auxiliary parameter (7), if it is set equal to zero,

\[ \omega_{0\mathrm{H},\mathrm{B}}^* = \sqrt{ \frac{ x_c(x_{13}+x_m+x_e)-r_{13}^2 \pm \sqrt{\left[x_c(x_{13}+x_m+x_e)-r_{13}^2\right]^2 -4x_c^2 x_e(x_{13}+x_m)} } {2x_e(x_{13}+x_m)} } \tag{10} \]

(the minus sign before the radical is used in determining \(\omega_{0\mathrm{H}}^*\)).

Expressions for the limiting angular rotor speeds can also be obtained using (4), (9), and (10). The rotor speeds at which self-excitation occurs, calculated and measured from oscillograms, differ too greatly (by up to 30%), which is explained by the assumptions adopted.

Self-excitation of the motor can be eliminated either by merging the boundaries of the resonance zone (or, at least, by bringing them sufficiently close together so that during acceleration the motor passes through this zone by inertia), or by raising the lower boundary beyond the limits of the operating speeds of the motor. From equality (10) it follows that

in principle any parameter of the system affects the value of the boundary frequencies of self-excitation, but, if the designs of the tripler and the motor have already been completed, in practice it is expedient and convenient to vary only the capacitance of the capacitors and the inductance of the load.

Putting \(\omega_{\mathrm{н}}^{*}=\omega_{\mathrm{в}}^{*}\) (coalescence of the boundaries of the resonance zone), we obtain, on the basis of (10), the condition for the absence of self-excitation

\[ x_c= \frac{r_{13}^{2}\left[x_{12}+x_m+x_{\mathrm{э}}+2\sqrt{x_{\mathrm{э}}(x_{13}+x_m)}\right]} {(x_{13}+x_m-x_{\mathrm{э}})^2}. \tag{11} \]

Fig. 3.

Analysis of expression (11) shows that the strongest influence on the self-excitation process is exerted by the capacitance, the active resistance of the stator branch, and the motor inductances. Elimination of self-excitation by increasing the active resistance of the stator branch or by decreasing the resistance of the capacitors is associated with considerable energy losses, deterioration of the starting characteristics of the motor, and of the weight indices of the tripler (for example, for our installation, according to calculation, \(r_{13}\simeq 9.8\ \Omega\) is required with a normal capacitance of \(218\,\mu F\), or \(C=2100\,\mu F\), if the active resistance of the circuit is not increased artificially).

Electric motors with a large magnetizing current (smaller \(x_m\)), other conditions being equal, are less prone to self-excitation, since they have a narrower zone of unstable operation. However, the parameters of \(150\) cycles/sec motors are determined by their operating and starting characteristics and, for some motors, are quite

definite. Therefore, in those cases when the motor is subject to self-excitation, the latter can be detuned by connecting, in parallel with the motor, directly beyond the capacitors, chokes or another motor (Fig. 4), which reduce the equivalent inductive resistance of the circuit. Figure 1,a presents the mechanical characteristic of the motor during no-load starting and with three single-phase chokes connected in parallel; Fig. 1,b shows the motor leaving the resonance zone after the same chokes are connected.

The power of the chokes is selected from the condition of ensuring normal starting of the least powerful motor among the motors supplied from the given tripler.

On the basis of expression (11), the required power of the starting chokes can be calculated. In the experiments, the power of the chokes was 25–30% of the connected power of the motor; satisfactory starting and operating characteristics of the motor were obtained. The use of chokes adjustable by bias magnetization promises to improve the motor-starting conditions and at the same time to increase the energy indices of the installation. After starting is completed, the chokes may be disconnected. Three-phase chokes are less desirable, since they create an unbalanced load and distort the shape of the voltage and current curves of the tripler.

Fig. 4.

The main drawback of chokes—a reduction in the power factor of the installation—can be eliminated by a slight increase in the power of the capacitor bank. It is preferable, instead of chokes, to connect in advance a non-self-exciting motor in those cases where this is acceptable.

Another method of eliminating the generator mode of the motor was noted above: raising the lower boundary of the resonance zone beyond the operating speeds of the motor, practically up to the synchronous speed. This is achieved, other conditions being equal, by reducing the capacitance of the series-compensation capacitors. The value of the capacitive reactance at which self-excitation will not occur is obtained by equating the lower limiting frequency of the free oscillations to the angular frequency at the output of the tripler, i.e., by taking in (10) \(\omega_0^2 = 1\):

\[ x_c = \frac{1}{2} \left( x_{13} + x_m + x_3 + \sqrt{ \left( \frac{x_m}{\dfrac{x_3}{x_m}+1} \right)^2 -4r_{13}^2 } \right). \tag{12} \]

Neglecting the relatively small quantities \(\dfrac{x_3^2}{x_m}\) and \(4r_{13}^2\), we finally obtain

\[ x_c \simeq x_{13}+x_m+ \frac{1}{2}\, \frac{x_3 x_m}{x_3+x_m} \simeq x_3+x_1+x_m+\frac{1}{2}x_2'. \tag{13} \]

Thus, the capacitors must compensate practically for the inductance of the circuit when the motor is idling, so that self-excitation is impossible. But such compensation is unsuitable for the starting and loaded operating conditions of the motor, since in these cases a considerably smaller capacitor reactance is required,

\[ x_c=x_i+\frac{U_0}{\sqrt{3}I_g} \left(U_g^*\sin\varphi_g-\sqrt{1-\left(U_g^*\cos\varphi_g\right)^2}\right) \tag{14} \]

(neglecting the resistances of the supply line, tripler, and motor), where

\(U_0\) is the no-load voltage of the tripler;

\[ U_g^*=\frac{U_g}{U_0} \]

is the relative value of the motor voltage;

\(I_g,\ \cos\varphi_g\) are the current and power factor of the motor.

The conditions (13) and (14) can be reconciled only by changing the capacitance during starting [2]: in the initial starting period the capacitance value corresponds to condition (14); as the starting current decreases, the capacitive reactance must be increased in accordance with (13); after starting is completed it is expedient to reduce the reactance of the capacitance to the value (14). To carry out the indicated program of capacitance variation, a rather complex control circuit is required, the advisability of which is doubtful. In a simplified form, a smooth reduction of the capacitance as a function of the starting current, with simultaneous regulation of the voltage on the motor, is achieved by shunting the capacitors with saturating reactors, in which the magnetomotive forces of the exciting winding and of the motor-current negative-feedback winding are directed oppositely. During short circuits beyond the capacitors, the reactors saturate and shunt the capacitors to a great extent, protecting them against overvoltages (Fig. 5).

[Diagram]

Fig. 5.

In the experimental installation, single-phase reactors without premagnetization were connected in parallel with the capacitors. The damping of free oscillations and the protection of the capacitors against overvoltages during short circuits were satisfactory. The total rating of the reactors did not exceed 20–25% of the connected motor rating. In the initial starting period, because of saturation, the reactors shunted the capacitors excessively, and the starting characteristics of the motor deteriorated. It is therefore necessary to use reactors with premagnetizing windings.

Reactors connected in parallel with the capacitors impart to the tripler essentially the same disadvantages as those connected in parallel with the motor.

Premagnetization of the tripler by direct current also raises the lower limiting frequency of free oscillations, but only slightly, and self-excitation is not detuned, since the influence of the inductive reactance of the tripler on the self-excitation process is small and the reactance changes within relatively narrow limits.

In the same direction, but more effectively, acts an increase in the input voltage of the tripler during the process of starting the motor; however, the bulkiness of the device for regulating the input voltage does not permit this method to be used.

In a number of cases self-excitation is not observed in the absence of additional devices, while at the same time acceptable starting and operating characteristics of the motor are ensured. Judging from the experiments on always normal starting of the VChS-10 electric drill from a compensated tripler and on the basis of expressions (13) and (14), it may tentatively be noted that a favorable combination of installation parameters is one in which the inductive reactance of the motor at no load, the tripler, and the resistance of the capacitors are of approximately the same order.

Conclusions

1. In the system static tripler—series-compensation capacitors—asynchronous motor, with an unfavorable ratio of the parameters, self-excitation of the motor occurs.

2. It is most desirable to eliminate self-excitation by selecting such parameters of the tripler and capacitors that even the least powerful motors are not subjected to self-excitation during starting and, at the same time, acceptable starting and operating characteristics of the motors are ensured. Equalities (11), (13), and (14) may serve as an approximate guide for assigning such parameters.

3. If it is inadvisable or impossible to eliminate self-excitation without the use of special means, reactors connected in parallel with the motor or with the capacitors should be used.

References

1. C. F. Wagner. Self-excitation of induction motors with series capacitors. Trans. AIEE V. 60, 1941.

2. V. T. Zagorskii. Investigation of the starting of squirrel-cage asynchronous motors from generators of comparable power, 1958.

3. V. G. Bauman, O. V. Ivanov, B. I. Komarov. Self-excitation of asynchronous motors with series capacitors. Elektrichestvo, No. 5, 1961.

4. A. T. Golovan, Chan Chan-Shen. Mechanical hunting of an asynchronous machine during its operation with series-connected capacitors. Elektrichestvo, No. 10, 1962.

5. A. A. Bessonov. Theoretical Foundations of Electrical Engineering. Gosenergoizdat, 1961.

6. A. I. Zaitsev, M. P. Tabinskii, A. S. Bobrovskii. Recording the mechanical characteristics of an asynchronous motor by measuring the speed and acceleration of the motor in transient regimes. Proceedings of the Tomsk Polytechnic Institute, vol. 117, 1963.

Received by the editors
in June 1962.