DETERMINATION OF OSCILLATION PARAMETERS IN DC ELECTRIC DRIVES WITH PULSE SPEED CONTROL

V. A. Beinarovich

(Presented by the scientific seminar of the Electromechanical Faculty)

Recently, discrete automatic-control systems have been finding ever wider application in engineering. As applied to the control of electric drives, such systems are usually constructed on the principle of regulating the power supplied to the motor by relay or pulse quantization of the energy from a constant-voltage source, or by periodically changing the parameters of the system. In modern electric drives, contactless devices based on semiconductor, ionic, and magnetic amplifiers are used as relay and pulse elements. Especially promising is the use of silicon controlled diodes—thyristors—which make it possible to create highly efficient DC and AC electric drives.

Thyristors and ionic devices, as is known, belong to discrete valves with limited control, in which the control signal determines only the instant at which current begins to flow through the valve, while the cessation of current is determined by the anode voltage of the valve. In this case the motor current may be discontinuous or continuous in character, and the study of transient conditions of the electric drive in the general case constitutes a complicated discrete nonlinear transcendental problem.

The study of steady-state and transient operating conditions of electric drives with ionic valves under continuous control of the firing angle has been carried out quite fully in the works of I. L. Kaganov [1], A. A. Bulgakov [2], A. V. Posse, Förster, and a number of other authors. The results of these studies are also applicable to thyristor electric drives with firing-phase control. However, only a few works by M. G. Chilikin, D. P. Morozov, and L. M. Tverdin [3] are devoted to the study of drives with valves of limited control under relay and pulse regulation. The possibilities of such systems have been little studied, and this hinders their spread in practice. At the same time, electric drives with relay and pulse contactless regulation are characterized by simplicity of design, low installed power, high dynamic qualities, noncriticality to the constancy of the characteristics and parameters of the system, and good energy

indicators. The disadvantages of such systems include more severe operating conditions for the motors (with respect to heating and commutation) and the presence of speed pulsations. In works [4, 5], investigations were carried out of the operating modes of the motor in thyristor electric drives with pulse regulation, and an engineering method was proposed for selecting the motor according to power. The present article considers the questions of determining the maximum amplitude and frequency of the motor-speed pulsations and the influence of the elastic couplings of working mechanisms.

In thyristor electric drives with separately excited motors, the speed pulsations have their greatest values under relay control of the converter grid circuit, “on—off,” implemented according to a single-valve (single-pulse) block diagram (Fig. 1). Owing to the limited controllability of the discrete valve and the mode of discontinuous currents, the pulsation amplitude cannot be reduced

Fig. 1

to zero by any increase in the gain of the regulator and its speed of response. Let us consider this most unfavorable case under the following assumptions: the regulator has an ideal relay characteristic; the motor excitation flux is constant; the inductance \(L\) and resistance \(R\) of the armature circuit are unchanged; armature reaction, hysteresis, eddy currents, and the valve turn-on and turn-off times are not taken into account.

For the interval of the current pulse (Fig. 2), the motion of the drive is described by the equations of electrical and mechanical equilibrium (in relative units):

\[ \sin(\omega t+\varphi_{з})-\nu=\varepsilon+i+T\frac{di}{dt}; \tag{1} \]

\[ i=i_c+B\frac{d\nu}{dt}; \tag{2} \]

where

\[ \nu=\frac{c_e}{U_m}n \]
is the motor speed (1);

\[ \varepsilon=\frac{\Delta E}{U_m} \]
is the voltage drop in the valve (1);

\[ i=\frac{R}{U_m}I \]
is the motor armature current (1);

\[ T=\frac{L}{R} \]
is the electromagnetic time constant of the armature circuit (sec.);

\[ i_c=\frac{R}{c_m U_m}M_c \]
— the static-load current (1);

\[ B=\frac{GD^2R}{375\,c_e c_m} \]
— the electromechanical time constant of the drive (sec);

\(U_m\) — amplitude of the supply voltage (V);

\(\omega\) — angular frequency of the supply voltage \(\left(\dfrac{\text{rad}}{\text{sec}}\right)\);

\(\varphi_z\) — valve firing angle (rad).

Fig. 2

The greatest upward deviation of the speed from the prescribed mean value corresponds, obviously (Fig. 2), to the application of an “excess” current pulse at the moment when the motor speed is close, practically equal, to the prescribed mean value. The exact solution of the equilibrium equations (1, 2) is too cumbersome. The speed deviation \(\Delta v_1\) during the flow time \(\tau_1\) of the current pulse can be found on the basis of the impulse theorem from the expression

\[ \Delta v_1=\frac{1}{B}\int_0^{\tau_1}(i-i_c)\,dt =\frac{\tau_1}{B}(i_1-i_{c1}), \tag{3} \]

where \(i_1\) is the mean value of the current during the pulse \(\tau_1\),
\(i_{c1}\) is the mean value of the load current during the pulse \(\tau_1\).

It follows from (3) that the speed deviation during the pulse does not depend on the current waveform and is determined only by the mean values of the pulse current and the load current.

For the pause interval the current is absent and the motion is described by the equation

\[ i_c+B\frac{dv}{dt}=0. \tag{4} \]

The greatest downward deviation of speed \(\Delta \nu_2\) from the prescribed mean value \(\nu_0\) under the action of the static load corresponds to the case when, at the beginning of the pause, the speed is close to the prescribed value (Fig. 2). Analogously to (3), we have

\[ \Delta \nu_2=\frac{1}{B}\int_0^{\tau_2} i_c\,dt=\frac{\tau_2}{B} i_{c2}, \tag{5} \]

where \(i_{c2}\) is the mean value of the load current during the pause \(\tau_2\).

On the basis of (3) and (5), the amplitude of the speed pulsations is determined as

\[ \Delta \nu=\Delta \nu_1+\Delta \nu_2 =\frac{\tau_1}{B}\left(i_1-i_{c1}\right)+\frac{\tau_2}{B}i_{c2}. \tag{6} \]

In practical calculations it is more convenient to take into account the rectified current averaged over a period, \(i_{\mathrm{cp}}\) (for the uncontrolled-rectifier mode), regarding it as a function of the prescribed mean speed value \(\nu_0\). These dependences are well known and are usually presented in the form of graphs convenient for use [1, 4, 5]. When determining \(i_{\mathrm{cp}}\) in this way, the influence of the speed deviation from \(\nu_0\) on the current is not taken into account. However, such an inaccuracy has little effect on the calculation results, owing to the insignificant nonlinearity of the speed increase over a half-period interval, since at industrial frequency the duration of the half-period is many times smaller than the electromechanical time constant.

Taking into account the mean rectified current,

\[ i_{\mathrm{cp}}\frac{2\pi}{\omega}=i_1\tau_1=i_1\frac{\lambda}{\omega}. \tag{7} \]

The pause is

\[ \tau_2=\frac{2\pi-\lambda}{\omega} \tag{8} \]

and the maximum amplitude of the pulsations at constant load \(i_c=\mathrm{const}\) is determined as

\[ \Delta \nu=i_{\mathrm{cp}}\frac{2\pi}{\omega B} +2i_c\frac{\pi-\lambda}{\omega B}, \tag{9} \]

where \(\lambda\) is the current-flow angle [1, 4].

The period of the speed oscillation at constant load can be determined as the sum of the pulse interval and the time of speed decrease by the value of the pulsation amplitude \(\Delta \nu\) under the action of the load:

\[ T_k=\frac{\lambda}{\omega}+\Delta \nu\,\frac{B}{i_c} =\frac{1}{\omega}\left[2\pi\left(1+\frac{i_{\mathrm{cp}}}{i_c}\right)-\lambda\right]. \tag{10} \]

The lowest pulsation frequency is

\[ f_k=\frac{1}{T_k}. \tag{11} \]

Knowing the parameters of the speed oscillations and its mean value, one can determine, from the mechanical characteristics of the drive, the mean value of the current and the maximum amplitude of the current pulsations.

When there are elastic couplings in the kinematic chain machine–tool–workpiece (MTW), for example in feed drives of metal-cutting machine tools, the elastic links play the role of low-frequency filters. The ampli-

the amplitude of the speed pulsations at the output of the elastic elements is determined from the expression

\[ \Delta \nu_c = \frac{\Delta \nu}{\sqrt{1 + (2\pi f_k T_c)^2}}, \tag{12} \]

where \(T_c\) is the time constant of the elastic element (machine tool).

The relations obtained make it possible, from the characteristics of an uncontrolled rectifier, for a circuit with known parameters and at a specified average speed, to determine the amplitude and frequency of the pulsations of the speed and current of an independently excited motor in circuits with an ideal relay controller and discrete valves with limited control (ionic devices or thyristors).

LITERATURE

1. I. L. Kaganov. Electronic and Ionic Converters. GEI, 1956.

2. A. A. Bulgakov. Fundamentals of the Dynamics of Controlled Valve Systems. Academy of Sciences of the USSR, 1963.

3. M. G. Chilikin, D. P. Morozov, L. M. Tverdin. Pulse control of the rotational speed of DC motors. In the collection Electric Drive and Automation of Industrial Installations. GEI, 1960.

4. V. A. Beinarovich, Yu. M. Achkasov, A. I. Zaitsev. The use of low-power motors in ionic electric drives with pulse control of rotational speed. Izvestiya TPI, vol. 117, 1963.

5. Yu. M. Achkasov, V. A. Beinarovich, A. I. Zaitsev. Checking a motor for heating in a medium-power ionic drive with pulse control of rotational speed. Izvestiya TPI, vol. 117, 1963.