SOME PROBLEMS IN THE THEORY OF SLIP-RING INDUCTION MOTORS UNDER PULSE SPEED CONTROL

Yu. P. KOSTYUKOV

(Recommended by the scientific seminar of the Electromechanical Faculty)

In existing control systems for slip-ring induction motors, speed regulation is carried out by introducing an additional resistance into the rotor circuit and by cascade schemes.

With the advent of powerful semiconductor uncontrolled and controlled valves, cascade schemes have entered a new phase of their development.

In some cases, it may prove expedient to use the pulse method for regulating the speed of slip-ring induction motors.

The basic circuits of pulse regulation have been described in sufficient detail in the domestic [1] and foreign literature.

Regulation of the rotational speed in such circuits is effected by changing the relative duration of conduction of the power thyristor, which is equal to

\[ \gamma = \frac{T - t_{\mathrm{и}}}{T}, \tag{1} \]

where: \(T\) is the pulse repetition period;
\(t_{\mathrm{и}}\) is the pulse duration, corresponding to the time during which the converter (or the additional resistance) conducts current.

The accuracy of calculating the static characteristics and determining the energy indices of pulse systems depends to a considerable degree on the harmonic analysis of the rotor phase current.

Harmonic analysis of the rotor phase current. We shall assume that the motor operates in steady state, the rotor winding is connected in star, and rectification is performed by means of semiconductor valves connected in a three-phase bridge circuit; in the rectified-current circuits the presence of a smoothing reactor with finite inductance is assumed.

Experimental investigation of commutation at various rectifier loads shows that the variation of the currents in these sections ...

as in practice occurs according to a linear law. This is in full agreement with the theory of valve devices, since in our case, especially at small slips, the active resistance of the rotor has a significant influence on the commutation process of the uncontrolled valves. With pulse speed control, in the interval between commutation periods, current pulsations arise in the rotor circuit, whose magnitude depends on the inductance of the intermediate circuits, the pulse repetition period, etc. The current pulsations can also be approximated by straight-line segments. Under these conditions, the phase current of the rotor can, with a sufficient degree of accuracy, be represented in the form of an “internal” component with height \(I_1\) and a “pulsating” component with height \(I_2-I_1\). Such an approximation reflects real processes with good accuracy in systems with a variable commutation frequency \((t_u=\mathrm{const})\) at large \(\gamma\), even for circuits without a smoothing reactor. In Fig. 1 the valve overlap angle \(\lambda\), and the parameters of the pulse device \(T'\), \(t'_u\), are referred to a stationary rotor; \(I_1\) and \(I_2\) are, respectively, the minimum and maximum values of the current of the established periodic process. Current pulsations are determined by the method set forth in [2].

Fig. 1

Let also the number of periods \(T'\) in the interval \(\dfrac{2\pi}{3}-\lambda\) be equal to \(n\), where \(n\) is an integer, i.e.

\[ n=\frac{\dfrac{2\pi}{3}-\lambda}{T'} . \tag{2} \]

For the “internal” component we choose the origin of coordinates in the middle of the pause of the phase current (point O). The curve under consideration is symmetric with respect to the abscissa axis and the origin; it is expanded into a Fourier series that contains no constant component, and no even or cosine harmonics. Applying the usual methods for expanding symmetric functions into a Fourier series [3, 4], we finally obtain the expression for the sine coefficient of the harmonic of order \(k\):

\[ b_k=\frac{4I_1\sin k\frac{\lambda}{2}}{\pi\lambda k^2} \left(\cos k\frac{\pi}{6}-\cos k\frac{5\pi}{6}\right). \tag{3} \]

Obviously, in the expansion of the “internal” component there will be no harmonics that are multiples of three, and the series will take the following form:

\[ f(x)=\frac{4\sqrt{3}I_1}{\pi\lambda} \left( \sin\frac{\lambda}{2}\cdot\sin x -\frac{1}{5^2}\sin\frac{5}{2}\lambda\cdot\sin 5x -\frac{1}{7^2}\sin\frac{7}{2}\lambda\cdot\sin 7x+\cdots \right). \tag{4} \]

For the expansion of the “pulsating” component, we shift the latter to the abscissa axis and place the origin at the same point O. This curve is symmetric with respect to the \(X\) axis and is expanded into a Fourier series that contains no constant term and no even-order harmonics.

For the sine coefficient of the harmonic of order \(k\), the following expression was obtained:

\[ \begin{aligned} B_k={}&\frac{2(I_2-I_1)}{\pi \gamma T' \cdot k^2} \sum_{m=1}^{m=n}\left[ \sin k\left(\frac{\pi}{6}+\frac{\lambda}{2}+mT'-t'_i\right) -\right.\\ &\left. -\sin k\left(\frac{\pi}{6}+\frac{\lambda}{2}+mT'-T'\right) \right]+\\ &+\frac{2(I_2-I_1)}{\pi t'_i \cdot k^2} \sum_{m=1}^{m=n}\left[ \sin k\left(\frac{\pi}{6}+\frac{\lambda}{2}+mT'-t'_i\right) -\right.\\ &\left. -\sin k\left(\frac{\pi}{6}+\frac{\lambda}{2}+mT'\right) \right]. \end{aligned} \tag{5} \]

In analogous fashion we obtain for the cosine coefficient

\[ \begin{aligned} C_k={}&\frac{2(I_2-I_1)}{\pi \gamma T' \cdot k^2} \sum_{m=1}^{m=n}\left[ \cos k\left(\frac{\pi}{6}+\frac{\lambda}{2}+mT'-T'\right) -\right.\\ &\left. -\cos k\left(\frac{\pi}{6}+\frac{\lambda}{2}+mT'-t'_i\right) \right]+\\ &+\frac{2(I_2-I_1)}{\pi t'_i \cdot k^2} \sum_{m=1}^{m=n}\left[ \cos k\left(\frac{\pi}{2}+\frac{\gamma}{2}+mT'\right) -\right.\\ &\left. -\cos k\left(\frac{\pi}{6}+\frac{\lambda}{2}+mT'-t'_i\right) \right]. \end{aligned} \tag{6} \]

The first harmonic of the “pulsating” component is

\[ i_{\mathrm{p}}=A_1\sin(x+\psi_1), \tag{7} \]

where

\[ A_1=\sqrt{B_1^2+C_1^2}\quad \text{and}\quad \psi_1=\operatorname{arctg}\frac{C_1}{B_1}. \tag{8} \]

After simple trigonometric transformations, the expression for the amplitude of the first harmonic of the “pulsating” component assumes the following form:

\[ A_1=\frac{(I_2-I_1)T'}{\pi} \sqrt{ \left(\sum_{m=1}^{m=n}\sin mT'\right)^2+ \left(\sum_{m=1}^{m=n}\cos mT'\right)^2 }. \tag{9} \]

The last expression was obtained under the condition

\[ \sin\frac{\gamma T'}{2}\simeq \frac{\gamma T'}{2},\quad \sin\frac{t'_i}{2}\simeq \frac{t'_i}{2},\quad \sin\frac{T'}{4}\simeq \frac{T'}{4}, \]

which is quite valid for \(n \ge 3\).

From the known first harmonics of the “pulsating” and “internal” components, one can determine the amplitude of the first harmonic of the rotor phase current and its phase relative to the adopted origin of coordinates.

As the corresponding calculations show, for \(\dfrac{I_2}{I_1}<1.5\) one may take \(\Psi_1=0\); in this case the amplitude of the first harmonic of the rotor phase current is determined with accuracy to the third—fourth decimal place, and the error in determining its phase is less than one electrical degree. Thus, the shift of the fundamental harmonic of the rotor phase current relative to the e.m.f. of the asynchronous motor is approximately equal to \(\dfrac{\lambda}{2}\), i.e., it is determined in the same way as is usually accepted in the theory of rectifier devices, and the effective value of the first harmonic of the rotor phase current is determined by the expression

\[ i_1=\frac{I_1}{\pi\sqrt{2}}\left[ \frac{4\sqrt{3}\sin\frac{\lambda}{2}}{\lambda}+ +\left(\frac{I_2}{I_1}-1\right)T'\sqrt{ \left(\sum_{m=1}^{m=n}\sin mT'\right)^2+ \left(\sum_{m=1}^{m=n}\cos mT'\right)^2 }\right]. \tag{10} \]

The harmonic analysis carried out for the rotor phase current makes it possible to assess the influence of the higher harmonics on the operation of the asynchronous machine.

Determination of the distortion coefficient. The distortion coefficient is defined as the ratio of the rms value of the fundamental harmonic to the rms value of the entire curve [5]:

\[ \nu=\frac{i_1}{i_{\mathrm{d}}}. \tag{11} \]

Let us determine the rms value \(i_{\mathrm{d}}\) of the phase current, using the known relation for periodic currents [5]. Performing the integration and simplifying the transformations, we obtain

\[ i_{\mathrm{d}}=\frac{I_1}{\sqrt{3}}\sqrt{ \frac{2}{3}+\frac{\lambda}{\pi} +\frac{I_2}{I_1}\left(\frac{I_2}{I_1}+1\right) \left(\frac{2}{3}-\frac{\lambda}{\pi}\right) }. \tag{12} \]

On the basis of (10) and (12), the distortion coefficient is equal to

\[ \nu= \frac{ \sqrt{3}\left[ \frac{4\sqrt{3}\cdot \sin\frac{\lambda}{2}}{\lambda} +\left(\frac{I_2}{I_1}-1\right)T' \sqrt{ \left(\sum_{m=1}^{m=n}\sin mT'\right)^2+ \left(\sum_{m=1}^{m=n}\cos mT'\right)^2 } \right] }{ \pi\sqrt{2}\sqrt{ \frac{2}{3}+\frac{\lambda}{\pi} +\frac{I_2}{I_1}\left(\frac{I_2}{I_1}+1\right) \left(\frac{2}{3}-\frac{\lambda}{\pi}\right) } }. \tag{13} \]

At the boundary of transition from continuous current to intermittent current \(I_1=0\); since the intermittent-current mode is possible at small loads, then \(\lambda\approx0\), and

\[ \nu=\frac{3T'}{2\pi}\sqrt{ \left(\sum_{m=1}^{m=n}\sin mT'\right)^2+ \left(\sum_{m=1}^{m=n}\cos mT'\right)^2 }. \tag{14} \]

Calculations by formula (14) show that the distortion coefficient \(\nu\) at the boundary of transition from continuous current to intermittent current does not exceed 0.85.

Table 1 gives the dependence of the distortion coefficient \(\nu\) in the continuous-current mode on the valve overlap angle \(\lambda\), the number of periods \(n\), and the current ratio \(\frac{I_2}{I_1}\). The calculations were performed by formula (13).

It follows from Table 1 that, with small current pulsations \(\frac{I_2}{I_1}=1.1\div1.5\), the influence of the higher harmonics of the “pulsating” component will be insignificant; moreover, for given \(\frac{I_2}{I_1}\) the distortion coefficient increases as \(n\) decreases, i.e., as the pulse repetition period increases, which once again emphasizes the advantages of systems with variable commutation frequency [2].

Selection of the motor power. As a first approximation, it may be assumed that the heating of the motor is determined by the rms value of the current, while useful work is performed only by the first harmonic; then the motor utilization factor is equal to the distortion coefficient. In addition, it is necessary to take into account the reduction of the electromagnetic

Table 1

of the motor torque as a result of valve overlap. This reduction is determined by the relation

\[ \frac{M_{\mathrm{i}}}{M_{\mathrm{en}}} = \nu \frac{\cos \dfrac{\lambda}{2}}{\cos \psi_2}\frac{E_c}{E_2}, \tag{15} \]

where: $M_{\mathrm{en}}, M_{\mathrm{i}}$ are, respectively, the rated electromagnetic torque of the motor and the electromagnetic torque under pulse control;

$\psi_2$ is the angle of displacement between the secondary emf and the current of an ordinary induction motor.

This expression was obtained for equal rms values. Thus, the motor is selected by the usual methods of electric-drive theory, but with an overrating of the power according to expression (15). Under pulse control an induction motor is, on average, underutilized in power by 10%.

Circle diagram. The circle diagram of an induction motor is easy to construct if the “pulsating” component of the rotor phase current is neglected. In this case we assume that the load current in the primary circuit has a sinusoidal form corresponding to the first harmonic of the “internal” component (with allowance for the leakage coefficient).

The coordinates of the center and the radius of the circle for cascade circuits are determined in [6]. We assign a certain value of the load current and slip, find the first harmonic of the “internal” component, and determine the valve overlap angle by an approximate method, taking into account the ohmic and inductive resistance in the commutating circuit [7]. Next we construct the circle diagram [6] and find the electromagnetic power $P_{\mathrm{e}}$. From the known electromagnetic power $P_{\mathrm{e}}$ and the specified slip $s$, we determine the relative on-duration that must be set in order to obtain precisely this slip. In the general case, the electrical power developed in the secondary circuit is equal to the electromagnetic power multiplied by the slip [8].

Thus, for cascade circuits with pulse control.

\[ P_{\mathrm{e}}\cdot s = E_{\mathrm{i}}\cdot I_1(1-\gamma) + I_1^2 R_{\mathrm{i}}(1-\gamma) + \Delta U I_1 + 3R_2\cdot i_1^2, \tag{16} \]

where: \(E_{\mathrm{i}}\) is the counter-emf of the converter;
\(R_{\mathrm{i}}\) is the equivalent ohmic resistance of the converter;
\(\Delta U\) is the voltage drop across the two uncontrolled bridge valves and across the protective diode (power thyristor) [2];
\(R_2\) is the active resistance of a phase of the rotor winding.

Expression (16) has been obtained under the condition that the current in the converter circuit and the emf of the converter device introduced into the rotor circuit have the form of rectangular pulses. From expression (16),

\[ \gamma = 1- \frac{P_{\mathrm{e}}\cdot s-\Delta U I_1-3R_2 i_1^2} {E_{\mathrm{i}} I_1+I_1^2 R_{\mathrm{i}}}. \tag{17} \]

The power factor obtained from the circle diagram must be multiplied by the distortion factor. The experimental results confirmed the correctness of the basic propositions set forth in this article.

REFERENCES

1. A. I. Zaitsev, Yu. P. Kostyukov. Pulse systems for speed control of asynchronous motors with a phase-wound rotor. In the present collection.

2. A. I. Zaitsev, Yu. P. Kostyukov. Asynchronous valve cascade with pulse control. Izvestiya TPI, 1965, vol. 153.

3. M. G. Serebrennikov. Harmonic Analysis. Gostekhizdat, 1948.

4. G. P. Tolstov. Fourier Series. Fizmatgiz, 1960.

5. G. V. Zeveke et al. Fundamentals of Circuit Theory. GEI, 1963.

6. D. A. Zavallishin et al. Regulation of the rotational speed of high-power asynchronous electric motors in a cascade circuit with a semiconductor converter. Izvestiya AN SSSR, OTN. Energetics and Automation, 1962, No. 3.

7. A. A. Bulgakov. Fundamentals of the Dynamics of Controlled Valve Systems. Publishing House of the Academy of Sciences of the USSR, 1963.

8. M. P. Kostenko and L. M. Piotrovsky. Electrical Machines, part 2, GEI, 1958.