On the Dynamics of an Electric Drive under Pulse Speed Control
Abstract Generated abstract
The paper analyzes the dynamics of a separately excited direct-current motor whose armature is supplied by intermittent rectangular voltage pulses, as may occur with controlled rectifier drives. Using standard acceleration and braking equations for a constant resisting torque, the authors approximate small speed deviations by linear segments and derive relations for pulse duration, pause duration, duty ratio, acceleration, deceleration, and dynamic speed variation. The analysis shows that the drive operates through alternating acceleration and deceleration intervals, with deceleration determined by load torque and inertia, while acceleration depends on the switching duty ratio. It further indicates that speed pulsation increases at low commanded speeds when the duty ratio is small, thereby limiting the lower range of pulse speed control.
Full Text
ON THE DYNAMICS OF AN ELECTRIC DRIVE UNDER PULSE SPEED CONTROL
A. P. ZAITSEV, A. I. ZAITSEV
(Presented by the scientific seminar of the Electromechanical Faculty)
The pulsed operating mode of a direct-current motor with independent excitation considered in this work is characterized by the fact that the voltage applied to the armature has an intermittent (pulsed) character. For drives of medium and high power, a controlled ionic rectifier can serve as the generator of pulsed voltage. With multiphase rectification it may be approximately assumed that the pulses of the rectified voltage have a rectangular form, which considerably simplifies the analysis of drive operation.
In view of the fact that a pulsed voltage is applied to the armature, the rotational speed of the motor has an oscillatory character. In Fig. 1 are shown the acceleration curve $\omega_p(t)$ and the straight braking line of the motor $\omega_t(t)$ for the case of constancy of the static resisting torque $M_c$ on the shaft. The equation of the motor acceleration curve has the form:
Fig. 1.
\[ \omega_p(t)=c_1 e^{\alpha_1 t}+c_2 e^{\alpha_2 t}+\omega_{\text{ust}}, \tag{1} \] [1]
where \(\omega_{\text{st}}\) is the steady-state value of the speed under load \(M_c\);
\(c_1, c_2\) are constants of integration, determined by the initial conditions;
\[ \alpha_{1,2}=-\frac{1}{2T_{\mathrm{я}}}\pm \frac{1}{2T_{\mathrm{я}}}\sqrt{1-\frac{4T_{\mathrm{я}}}{T_{\mathrm{м}}}} \]
are the roots of the characteristic equation;
\(T_{\mathrm{я}}\) is the electromagnetic time constant of the motor armature circuit;
\(T_{\mathrm{м}}\) is the mechanical time constant of the drive.
The equation of the straight line of braking is determined from the expression
\[ -M_c=J\frac{d\omega_m(t)}{dt}. \tag{2} \]
[1].
Separating the variables in (2) and taking into account the initial conditions in the integration, we obtain the desired equation
\[ \omega_m(t)=\omega_{\text{st}}-\frac{M_c}{J}t, \tag{3} \]
where \(t\) is the time measured from the instant at which braking begins;
\(M_c\) is the static resisting torque on the motor shaft;
\(J\) is the moment of inertia of the drive.
During operation in the pulse mode, the motor successively traverses sections of the acceleration curve and of straight-line braking, bounded by the width of the speed-oscillation zone \(\Delta\omega\). During application of a voltage pulse to the armature, the motor accelerates (section \(KM\)); during the pause it is braked under the action of the static resisting torque \(M_c\) (section \(MT\)).
For a sufficiently small deviation of the speed from the prescribed value, it may be assumed that the actual speed varies along straight lines parallel to the braking line and to the tangent to the acceleration curve at the point of intersection with the horizontal straight line \(\omega_{\text{set}}\) corresponding to the prescribed speed.
Introduce the following notation:
\(KN=t_{\mathrm{u}}\) is the time for which the voltage pulse is applied to the motor armature;
\(NT=t_{\mathrm{p}}\) is the pause time (absence of a pulse);
\(KT=KN+NT=t_{\mathrm{u}}+t_{\mathrm{p}}=t_0\) is the duration of the period of the voltage pulses;
\(MN=\Delta\omega\) is the dynamic speed difference (speed-oscillation zone);
\[ \frac{t_{\mathrm{u}}}{t_{\mathrm{u}}+t_{\mathrm{p}}} = \frac{t_{\mathrm{u}}}{t_0} = \varepsilon, \tag{4} \]
the relative duration of switching-on.
We carry out auxiliary constructions. To the motor acceleration curve at point \(P\) (the point of intersection of \(\omega_p(t)\) and \(\omega_{\text{set}}\)) we draw a tangent up to its intersection with the line \(\omega_{\text{st}}\). From the same point we draw a perpendicular to the line \(\omega_{\text{st}}\). The triangle \(ABP\) constructed in this way is similar to triangle \(KNM\), because of the parallelism of all sides. For the same reason, triangle \(OCE\) is similar to triangle \(NMT\). It follows from the foregoing that
\[ \frac{AP}{AB}=\frac{MN}{KN}, \tag{5} \]
and
\[ \frac{CO}{OE}=\frac{MN}{NT}. \tag{6} \]
The ratio
\[ \frac{AP}{AB}=\operatorname{tg}\gamma=\frac{d\omega_p(t)}{dt}=c_1\alpha_1 e^{\alpha_1 t}+c_2\alpha_2 e^{\alpha_2 t} \tag{7} \]
is determined as the derivative of the acceleration-curve equation at point \(O\). The segment \(OE\) represents the total braking time from \(\omega_m(t)=\omega_{\mathrm{ust}}\) to \(\omega_m(t)=0\) and is determined from (3)
\[ OE=t_{\mathrm{br}}=\frac{J\omega_{\mathrm{ust}}}{M_c}. \tag{8} \]
Substituting into (5) and (6) the corresponding values of the segments and ratios, we obtain
\[ \frac{d\omega_p(t)}{dt}=\frac{\Delta\omega}{t_i}, \tag{9} \]
\[ \frac{\omega_{\mathrm{ust}}}{t_{\mathrm{br}}}=\frac{\Delta\omega}{t_p}. \tag{10} \]
From (7), (9), and (10), the relative duration of switching on is
\[ \varepsilon=\frac{M_c}{M_c+J\left(c_1\alpha_1 e^{\alpha_1 t}+c_2\alpha_2 e^{\alpha_2 t}\right)}, \tag{11} \]
From (9) and (10), the drive acceleration during application of the voltage pulse is
\[ \tau_p=\left|\frac{\omega_{\mathrm{ust}}t_p}{t_{\mathrm{br}}t_i}\right| =\left|\frac{\omega_{\mathrm{ust}}(1-\varepsilon)}{t_{\mathrm{br}}\varepsilon}\right| =a_m\left|\frac{1-\varepsilon}{\varepsilon}\right|, \tag{12} \]
where
\[ a_m=\frac{\omega_{\mathrm{ust}}}{t_{\mathrm{br}}}=\frac{M_c}{J} \tag{13} \]
is the deceleration of the drive during the pause.
From (7), (9), (10), and (11) we determine the dynamic speed difference
\[ \Delta\omega=\frac{M_c}{J}\cdot \frac{1-\varepsilon}{\varepsilon}\cdot t_i, \tag{14} \]
or
\[ \Delta\omega=\frac{M_c}{J}\cdot t_p. \tag{15} \]
Conclusions
-
The drive operates in a continuous regime of accelerations and decelerations, which act alternately with a definite frequency depending on \(M_c\), \(J\), and \(\omega_{\mathrm{zad}}\). The magnitude of the acceleration depends on the relative duration of switching on \(\varepsilon\), while the magnitude of the deceleration is constant and depends only on \(M_c\) and \(J\).
-
The magnitude of the dynamic speed difference \(\Delta\omega\), characterizing the dynamic qualities of the drive, depends on \(M_c\), \(J\), and \(\varepsilon\) for an unchanged voltage-pulse time \(t_i\). Therefore, at a low speed \(\omega_{\mathrm{zad}}\), when the relative duration of switching on \(\varepsilon\) is small, a large speed pulsation is obtained, limiting the lower bound of speed regulation by the pulse method.
References
- A. T. Golovan. Fundamentals of Electric Drive. GEI, 1959.
Received by the editors
in May 1962.