ON THE COMPUTATIONAL DETERMINATION OF THE FREQUENCY OF NATURAL OSCILLATIONS AND THE INDUCTANCE OF DAMPING REACTORS IN THE SYSTEM FERROMAGNETIC FREQUENCY TRIPLER—SERIES-COMPENSATION CAPACITORS—INDUCTION MOTOR

Ya. V. PETROV, A. I. ZAITSEV

(Recommended by the scientific seminar of the Electromechanical Faculty).

Several methods are known for calculating the limiting conditions of self-excitation of induction motors connected through series-compensation capacitors. Some of these methods are based on analysis of the steady-state operating modes of an induction machine using Kirchhoff’s and Ohm’s laws for sinusoidal-current circuits, since at the stability boundary undamped natural oscillations are maintained in the system [1—6 et al.]. In a number of other works [1, 7—10 et al.], methods are used for studying the stability of automatic-control systems described by linear differential equations with constant coefficients; the state of the system at the stability boundary corresponds to the limiting conditions for self-excitation of the machine.

Natural oscillations in the system do not arise for an appropriate combination of its parameters. If, however, the system is subject to oscillations during acceleration of the motor, they can be eliminated by connecting, in parallel with the series-compensation capacitors or with the motor, ballast (damping) active or inductive resistances. Calculations and experiments have shown that, for installations up to 1—2 kW at a frequency of 150 Hz, reactors with a steel core are more effective dampers.

Although the cited works give fairly simple expressions for the lower and upper resonance frequencies, the discrepancy between calculated and experimental data is, as a rule, considerable, and the formula known to us for choosing the magnitude of the reactor inductance [8] gives too large a margin of reactor power in comparison with that required.

Without considering the complex transient electromechanical process of an induction machine undergoing self-excitation, the value of the motor self-acceleration speed and the corresponding frequency of natural oscillations can be established more accurately and simply on the basis of certain principles of the theory of electrical filters made of reactive elements, which are proposed here for appli-

changed. In this case, the choice of the inductance values of the reactors is also considerably simplified and refined.

Figures 1 and 2 show the equivalent circuits of the system, respectively, for the motor and generator modes of the asynchronous machine. Conventionally, it may be assumed that in the first case the electrical energy from the power source (the tripler), with e.m.f. \(e_3\) and internal parameters \(r_3\) and \(L_3\), is transmitted to the rotor circuit of the motor through the passive four-terminal network outlined by the dash-dotted contour. In the second case, under self-excitation, the electrical energy of the free oscillations, conversely, passes from the rotor circuit of the machine through the same four-terminal network to the source, being dissipated in the active resistances of the latter and of the four-terminal network. The active resistance of the rotor circuit in the generator mode of the machine is negative, generating, and, on the basis of the compensation theorem, may be replaced in the equivalent circuit by an equivalent e.m.f. \(e_{02}\) of the free oscillations, numerically equal to the voltage drop across this resistance and directed opposite to the current \(I'_{02}\) [11].

Fig. 1.

Fig. 2.

Thus, regardless of the machine mode, the electrical energy is transmitted through one and the same four-terminal network, in whose series arm are connected the capacitance of the series compensation, the active resistance and inductance of the line and of the stator circuit, while the parallel arm is formed by the active resistance and inductance of the magnetizing circuit of the machine.

We neglect the active resistances of the four-terminal network and consider the latter as an L-shaped band-pass filter of type K. The admissibility of freeing the four-terminal network from active resistances (a lossless filter) can be justified by the fact that, first, in the absence of a line or with a short line these resistances are comparatively small; second, active resistances, especially in the case of a long line of small cross-section, are taken into account by a corresponding reduction of the series-compensation capacitance in order to obtain the specified voltage at the motor, i.e., they affect the filter parameters in a favorable direction. In Fig. 3 the filter has only reactive resistances. In addition, we assume that the characteristic impedances of the filter are matched with the load impedances.

Since this filter is an L-shaped half-section of T- or \(\Pi\)-shaped symmetrical filters of type K, the pass condition for symmetrical filters consisting of reactive elements [12] is applicable to it:

\[ Z_1 \geq -4Z_2; \]
\[ Z_1 \leq 0, \tag{1} \]

where

\[ Z_1=2\left(\omega L_1-\frac{1}{\omega C}\right) \]

is the doubled equivalent resistance of the series branch;

\[ Z_2=\frac{1}{2}\omega L_m \]

is one half of the resistance of the parallel branch of the filter (Fig. 3).

Fig. 3.

On the basis of (1) we obtain the cutoff frequency of the upper frequencies

\[ f_{c1}=\frac{1}{2\pi\sqrt{C(L_1+L_m)}} \tag{2} \]

and the cutoff frequency of the lower frequencies

\[ f_{c2}=\frac{1}{2\pi\sqrt{CL_1}}. \tag{3} \]

The range \(f_{c1}\div f_{c2}\) bounds the passband (transparency band) of frequencies; outside this range lies the stopband (opacity band).

If free oscillations with frequencies lying in the transparency band can be generated in the system, self-excitation of the motor will occur, and the lower resonant frequency \(f_{\mathrm{on}}\) will be equal to the cutoff frequency \(f_{c1}\). If, however, the possible frequencies of the free oscillations \(f_{\mathrm{on}}\div f_{\mathrm{ov}}\) fall in the stopband, self-excitation does not arise.

The relation obtained, \(f_{\mathrm{on}}=f_{c1}\), coincides with the expression for the self-excitation frequency of an asynchronous generator according to [13], or, if the stator leakage inductance \(L_1\) is neglected, according to [3], where these dependences were established by another method.

The angular velocity of the rotor of the self-excited machine, in fractions of synchronous speed, is determined by the expression [14]:

\[ \omega_{2*}=\omega_{0*}\left\{1+ \frac{r'_2(r_1+r_2)} {\left(1+\dfrac{x'_2}{x_m}\right)^2 \left[ x_e^2\omega_{0*}^2+(r_1+r_3)^2-2x_e x_c+ \left(\dfrac{x_c}{\omega_{0*}}\right)^2 \right]} \right\}, \tag{4} \]

where \(\omega_{0*}\) is the angular frequency of the free oscillations in fractions of the supply-network frequency;

\[ x_e=x_1+x_3+\frac{x'_2x_m}{x'_2+x_m} \]

(the notation is as in Fig. 1).

The calculation of the self-excitation frequency according to \(f_{\mathrm{on}}=f_{c1}\) and of the corresponding critical speed from (4) is simple and agrees satisfactorily with experiment. Thus, for a 1.5 kVA tripler and a 0.5 kW motor (installation 1), the experimental self-excitation frequency* and critical speed are equal to:

\[ f_{\mathrm{on}}=78.1\ \text{Hz};\qquad n_{2h}=1640\ \text{rpm}. \]

* The frequencies of the free oscillations were determined by processing oscillograms of the speeds during self-run-up by the method described in [15].

The calculated values of these same quantities from (2) and (4) are:

\(f_{c1}=f_{\text{on}}=76\ \text{Hz}\) (error — 2.7%);

\(\omega_{2\text{n}*}=0.52\), or \(n_{2\text{n}}=1560\ \text{rpm}\) (error — 4.9%).

Neglecting the inductance \(L_1\) increases the accuracy of the calculation:

\(f_{c1}=f_{\text{on}}=78.5\ \text{Hz};\)

\(n_{2\text{n}}=1635\ \text{rpm}.\)

The discrepancy between the calculated and experimental frequencies and speeds for different degrees of compensation in the same installation does not exceed 8% (the errors are predominantly negative and increase with increasing degree of compensation).

Usually \(L_1 \ll L_m\), even when the inductance of the tripler–motor line is taken into account; therefore it is permissible to set \(L_1 \approx 0\). In this case we obtain a T-shaped high-frequency filter with cutoff frequency

\[ f_{c1}\approx \frac{1}{2\pi\sqrt{CL_m}}. \tag{2′} \]

For the same reason, the second cutoff frequency \(f_{c2}\) of the band-pass filter considerably exceeds the operating frequency \(f=150\ \text{Hz}\), and there is no need to determine the frequency \(f_{c2}\).

Investigation shows that, for experimental installations, as well as for installations with series compensation and other low-power three-phase asynchronous motors of industrial and elevated frequencies, the maximum self-excitation frequency depends little on the degree of compensation and on the variety of parameters of motors and power sources, and is \(f_{\text{ov}}=(0.7\div 0.8)f\). In view of the impossibility of complete matching of the load and characteristic impedances at different frequencies of the pass band, it is recommended for the band-pass filter and the high-frequency filter to take \(f_{c1}=0.7f\) [12]. Taking into account further that the active resistances of the system exert an additional damping effect and that the limiting frequency of free oscillations was calculated for synchronous speed, we obtain the simple condition for the absence of self-excitation:

\[ f_{c1}\geq 0.7f \tag{5} \]

or

\[ C(L_1+L_m)\leq \frac{0.0518}{f^2}. \tag{5′} \]

For \(f=150\ \text{Hz}\), we have

\[ f_{c1}\geq 105\ \text{Hz} \]

or

\[ C(L_1+L_m)\leq 2.3\cdot 10^{-6}. \]

For example, for installation I,

\[ f_{c1}=76\ \text{Hz}<105\ \text{Hz} \]

or

\[ 218\cdot 10^{-6}(1.4+18.9)\cdot 10^{-3}>2.3\cdot 10^{-6}, \]

that is, self-excitation is entirely possible, which is also observed in experiments.

For experimental installation II (tripler \(S_{\text{n}}=300\ \text{VA}\), motor \(P_{\text{n}}=150\ \text{W}\)), carried out with allowance for (5,5′), \(f_{c1}=114\ \text{Hz}>105\ \text{Hz}\), or

\[ 365\cdot 10^{-6}(0.265+5.1)\cdot 10^{-3}<2.3\cdot 10^{-6}, \]

that is, self-excitation should be absent—the experiment confirms this*.

* The \(D\)-partition method gives an erroneous result here: the calculation gives \(\omega_{\text{on}*}=0.67\) and \(\omega_{\text{ov}*}=0.845\); consequently, there is, although narrow, a self-excitation zone; however, during numerous motor starts no resonance phenomena of any kind were observed.

Expression (5) or (5′) makes it possible not only to answer quickly the question of whether self-excitation of the motor will occur in the given system, but also serves as one of the starting points in designing installations without self-excitation and in selecting damping reactors. If the motor parameters are specified, then, for the motor with the largest sum \(L_1+L_m\) (the line inductance is included in \(L_1\)), one should use (5′) to find the value of the series-compensation capacitance, and then, from the condition of creating the required voltage level at the motor terminals, calculate the tripler reactance \(X_3\), taking \(r_3 \le 0.1X_3\). If, conversely, the tripler parameters are known, the motor parameters are outlined analogously.

Fig. 4.

Figure 4 gives the equivalent circuit of the filter under consideration, with a reactor connected in parallel with the capacitors for suppressing self-excitation. A \(\Gamma\)-shaped blocking filter of type \(K\), or a high-pass filter of type \(m\), is obtained. It is convenient to determine the cutoff frequencies of the filter and the inductance of the reactor \(L_{\mathrm{dr}}\), taking \(L_1\) into account, by using condition (1),

\[ Z_1=2\left(\omega L_1-\frac{\omega L_{\mathrm{dr}}}{\omega^2 C L_{\mathrm{dr}}-1}\right); \qquad Z_2=\frac{1}{2}\omega L_m; \]

\[ \text{a) }\quad 2\left(\omega L_1-\frac{\omega L_{\mathrm{dr}}}{\omega^2 C L_{\mathrm{dr}}-1}\right) = -4\cdot \frac{1}{2}\omega L_m, \]

whence the cutoff frequencies are

\[ f_c=0; \]

\[ f_{c1}=\frac{1}{2\pi} \sqrt{\frac{L_{\mathrm{dr}}+L_1+L_m}{C L_{\mathrm{dr}}(L_1+L_m)}}. \tag{6} \]

\[ \text{b) }\quad 2\left(\omega L_1-\frac{\omega L_{\mathrm{dr}}}{\omega^2 C L_{\mathrm{dr}}-1}\right)=0 \]

and

\[ f_{c2}=\frac{1}{2\pi} \sqrt{\frac{L_{\mathrm{dr}}+L_1}{L_{\mathrm{dr}}L_1 C}}. \tag{7} \]

Here the pass band is \(f_{c1}\div f_{c2}\), the stop bands are \(0\div f_{c1}\) and \(f_{c2}\div \infty\), and the frequency of infinite attenuation is

\[ f_\infty=\frac{1}{2\pi\sqrt{L_{\mathrm{dr}}C}}. \]

As calculations show, the frequency \(f_\infty\) approaches the range of natural-oscillation frequencies in the system or lies within it, owing to which increased effectiveness is achieved in suppressing self-excitation by the reactor of the series circuit. The frequency \(f_{c2}\gg f\), and the former need not be determined.

From (6) we find the reactor inductance at which parametric resonance is eliminated during motor starting:

\[ L_{\mathrm{dr}}= \frac{L_1+L_m}{\omega_{c1}^{2}C(L_1+L_m)-1}, \tag{8} \]

where, as before, \(\omega_{c1}=2\pi f_{c1}\ge 2\pi\cdot 0.7f\).

Neglecting the relatively small inductance \(L_1\), we rewrite (6) and (8)

\[ f_{c1}\approx \frac{1}{2\pi}\sqrt{\frac{L_{\mathrm{dr}}+L_m}{C L_{\mathrm{dr}} L_m}}; \tag{6'} \]

\[ L_{\mathrm{dr}}\approx \frac{L_m}{\omega_c^2 C L_m-1}.^{*} \tag{8'} \]

Let us transform the inductive part of the filter with a shunt reactor (Fig. 5) into a form convenient for applying the filter pass condition. In the absence of load, the inductive part of the indicated circuit may be regarded as a Γ-shaped network with a parallel branch \(L_{\mathrm{dr}}\) and a series branch \((L_1+L_m)\). Passing to the equivalent T-shaped

Fig. 5.

Fig. 6.

equivalent circuit (Fig. 6) and using the method for transforming induction-motor equivalent circuits [16], we obtain:

\[ L_1''=\frac{L_1\cdot L_{\mathrm{dr}}}{L_1+L_{\mathrm{dr}}}; \]

\[ L_{\mathrm{dr}}''=\frac{L_{\mathrm{dr}}^2}{L_1+L_{\mathrm{dr}}}; \]

\[ L_{\mathrm{dr}}''=\frac{L_m}{\left(1+\dfrac{L_1}{L_{\mathrm{dr}}}\right)^2}. \tag{9} \]

The filter pass condition for the circuit of Fig. 6 gives:

\[ f_{c2}=\frac{1}{2\pi\sqrt{C L_1''}}; \]

\[ f_{c1}=\frac{1}{2\pi\sqrt{C\left(L_1''+L_e''\right)}}. \tag{10} \]

where

\[ L_e''=\frac{L_{\mathrm{dr}}''\cdot L_m''}{L_{\mathrm{dr}}''+L_m''} =\frac{L_m\cdot L_{\mathrm{dr}}^2}{(L_1+L_{\mathrm{dr}})(L_1+L_{\mathrm{dr}}+L_m)}. \tag{11} \]

* The same values of the cutoff frequency and the reactor inductance are obtained by considering the four-terminal network of Fig. 4 (without \(L_1\)) as a high-pass filter of type \(m\). The parameter \(m\) is calculated from the formula

\[ m=\frac{1}{\omega_{c1}\sqrt{2CL_m}}; \]

the reactor inductance is

\[ L_{\mathrm{dr}}=\frac{2m}{1-m^2}L_m; \]

the high-frequency cutoff frequency is

\[ f_{c1}=\frac{1}{2\pi m\sqrt{2CL_m}}. \]

Substituting \(L_1^{*}\) and \(L_e^{*}\) from (9) and (11) into (10), after simple transformations we find

\[ f_{c1}=\frac{1}{2\pi}\sqrt{\frac{L_{\mathrm{dr}}+L_1+L_m}{C L_{\mathrm{dr}}(L_1+L_m)}} \tag{12} \]

or

\[ L_{\mathrm{dr}}=\frac{L_1+L_m}{\omega_{c1}^{2} C (L_1+L_m)-1}. \tag{13} \]

But expressions (12) and (13) completely coincide with expressions (6) and (8), respectively; that is, regardless of the connection scheme of the given reactor (in parallel with the capacitors or with the motor—taking into account, in the latter case, the connection of the reactor windings in \(\lambda\)), the cutoff frequencies are the same, or, conversely, for a specified cutoff frequency \(f_{c1}\), the same value of the reactor inductance is required, at which self-excitation is not observed.

The validity of formula (13) has been confirmed experimentally. For one experimental installation, whose motor undergoes self-excitation, three single-phase reactors with windings connected in \(Y\) were made on the cores of current transformers 0—49у-100/5. By selecting the number of turns of the reactor windings, the following maximum inductance of the latter was obtained: \(L_{\mathrm{dr}}=24.7\cdot10^{-3}\) H, at which self-excitation did not occur during the starting of the motor from zero and nonzero initial speed. According to (13), however, the calculated inductance is

\[ L_{\mathrm{dr}}= \frac{(1.4+18.9)\cdot10^{-3}} {(2\pi\cdot105)^2\cdot218(1.4+18.9)\cdot10^{-9}-1} =21.9\cdot10^{-3}\ \text{H}, \]

that is, the error in determining the magnitude of the reactor inductance was \(-11.3\%\) (the latter decreases if the inductance \(L_1\) is neglected).

It may be asserted that expressions (8), (8′) are suitable for determining the value of the inductance of damping reactors connected in parallel with the capacitors or with the motor. With the aid of these same expressions it is possible to estimate the influence, on the self-excitation mode of the given motor, of other parallel-connected motors supplied jointly from one source.

For example, if the inductance of some motor operating in steady state, \(L_d\), satisfies (8) or (8′), that is, \(L_d \leqslant L_{\mathrm{dr}}\), then it should be expected that the starting of another motor, having parameters \(L_1\) and \(L_m\), will take place without self-excitation. In the opposite case, it will be necessary to use damping reactors with inductance

\[ L'_{\mathrm{dr}} \lesssim \frac{L_d \cdot L_{\mathrm{dr}}}{L_d - L_{\mathrm{dr}}}, \]

since without reactors the motor being started self-excites, the currents of free oscillations enter the running motor, a braking torque rapidly develops on the shaft of the latter, the motor overturns and becomes stuck at a certain intermediate speed, that is, it also self-excites.

Naturally, the greater the number of previously connected motors or other three-phase loads having a reactive component—in other words, the smaller the equivalent inductance of the other parallel-connected load—the less probable, according to (8), is self-excitation of the given motor during starting.

Parallel connection of an active three-phase load is not so effective: thus, for a 500 W motor of installation I it was necessary to switch in active resistances of 6.1 ohms per phase (power loss 1830 W) so that, when starting the motor from a nonzero initial speed, self-excitation would not be observed.

Conclusions

1. It is proposed to apply certain propositions of the theory of electrical filters made of reactive elements to the calculation of the principal quantities in a circuit with self-excitation of an induction motor.

2. Simple formulas have been obtained by means of which one can find the frequency and speed of the motor corresponding to the self-oscillation mode, and also select the value of the inductance of the damping reactor. The discrepancy between the calculated and experimental data lies within permissible limits.

References

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