Electromechanical Transient Processes in an Asynchronous Electric Drive under Pulse Control
Abstract Generated abstract
The paper investigates electromechanical transients in an asynchronous motor drive with pulse speed control through switched additional rotor resistance. Using differential equations for stator and rotor flux linkages in a synchronously rotating coordinate system, the authors construct an analog-computer model of the motor, pulse converter, and control circuit to study starting, load changes, inertia variation, and repeated switching. The results indicate that pulse control introduces electromagnetic torque pulsations at the commutation frequency, which decrease as this frequency rises and become relatively insignificant above about 1000 Hz, while reducing the relative on-time lowers the average torque and enables controlled starting with a prescribed dynamic characteristic. Experimental checks on an MT-112-6 motor are reported to agree with calculated torque values within about 10 to 15 percent.
Full Text
ELECTROMECHANICAL TRANSIENT PROCESSES IN AN ASYNCHRONOUS ELECTRIC DRIVE UNDER PULSE CONTROL
O. G. MAKARCHENKO, Yu. P. KOSTYUKOV
(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)
In electric drives with pulse control [1], operation of an asynchronous motor with frequent starts and braking is possible under intermittent duty, with application or removal of load and with changes in the relative duration of switching-on. Depending on the type of mechanism and the nature of the technological process, one or another transient mode is characteristic of these electric drives.
In studies of the dynamics of asynchronous electric drives, the experimental method and mathematical modeling based on the use of analog or digital computers have become most widespread.
In the present work, the dynamics of starting asynchronous motors with pulse speed control was investigated by the method of mathematical modeling on an analog computer.
Electromechanical transient processes in an asynchronous motor with a pulse converter and additional resistances in the rotor circuit of alternating current are described by the following system of differential equations:
$$ \frac{d\Psi_{\alpha s}}{d\tau} = U_{\alpha s} - \frac{r_1}{x'_s}\Psi_{\alpha s} + k_r\frac{r_1}{x'_s}\Psi_{\alpha r} + \Psi_{\beta s}; $$
$$ \frac{d\Psi_{\beta s}}{d\tau} = U_{\beta s} - \frac{r_1}{x'_s}\Psi_{\beta s} + k_r\frac{r_1}{x'_s}\Psi_{\beta r} - \Psi_{\alpha s}; $$
$$ \frac{d\Psi_{\alpha r}}{d\tau} = k_s\frac{r_2+bR_d}{x'_r}\Psi_{\alpha s} - \frac{r_2+bR_d}{x'_r}\Psi_{\alpha r} + (1-\omega)\Psi_{\beta r}; $$
$$ \frac{d\Psi_{\beta r}}{d\tau} = k_s\frac{r_2+bR_d}{x'_r}\Psi_{\beta s} - \frac{r_2+bR_d}{x'_r}\Psi_{\beta r} - (1-\omega)\Psi_{\alpha r}; \tag{1} $$
$$ \frac{d\omega}{d\tau} = \frac{k_r}{I\,2x'_s} \left( \Psi_{\beta s}\Psi_{\alpha r} - \Psi_{\alpha s}\Psi_{\beta r} \right) - \frac{M_c}{I\cdot p}, $$
where
\[
\Psi_{\alpha s},\ \Psi_{\beta s}
\]
— components of the generalized flux-linkage vector of the stator;
\[
\Psi_{\alpha r},\ \Psi_{\beta r}
\]
— components of the generalized flux-linkage vector of the rotor;
\[
U_{\alpha s},\ U_{\beta s}
\]
— components of the generalized voltage vector of the stator;
\(\omega\) — angular velocity of rotation of the rotor;
\(M_c\) — static load torque;
\[ k_s=\frac{x_m}{x_s};\quad k_r=\frac{x_m}{x_r} \]
— coupling coefficients of the stator and rotor;
\[ r_1;\quad x_s'=x_s-\frac{x_m^2}{x_r};\quad x_r'=x_r-\frac{x_m^2}{x_s};\quad x_s=x_1+x_m;\quad x_r=x_2'+x_m; \]
\(r_2\) — parameters of the T-shaped equivalent circuit of the asynchronous motor in relative units;
\(R_d\) — additional resistance of the rotor circuit;
\(b\) — logical coefficient taking the values 0, 1;
\(b=1\) when the thyristors of the pulse converter are switched off;
\(b=0\) when the thyristors are switched on.
Using the differential equations (1), written with respect to flux linkages in a synchronously rotating coordinate system, an electronic model was constructed of an asynchronous motor with a pulse converter and additional resistances in the rotor circuit. The conducting state of the valves of the pulse converter was simulated on the model by closing relay contacts. The control circuit of this relay was assembled on the modeling installation and consisted of a pulse generator and a pulse-width modulator.
The parameters of the asynchronous motor vary depending on slip and currents. In the pulse-control systems considered [2], the rotor current has a continuous character. At an increased commutation frequency, the content of higher harmonics in the motor current is negligibly small. Therefore, the influence of current displacement effects and saturation of the magnetic circuit in modeling an asynchronous motor can be taken into account by special methods [3].
The calculations performed for electromechanical transients in an asynchronous electric drive with pulse control showed that additional pulsations appear in the electromagnetic-torque curve, whose frequency is equal to the commutation frequency.
On the electronic model, the influence of the commutation frequency \(f_k\) on the magnitude and character of the electromagnetic torque of the asynchronous motor was investigated. The investigations showed that, as the commutation frequency increases, other conditions being equal, the pulsations of the electromagnetic torque caused by the influence of the pulse system decrease. Based on processing a large number of calculated oscillograms obtained on the model at nominal load and with variation of the commutation frequency, Fig. 1 plots the dependence of the pulsation coefficient of the electromagnetic torque \(M_2/M_1\) in the range of impact value on the commutation frequency. This dependence shows that, for \(f_k>1000\) Hz, the torque pulsation coefficient changes only slightly.
The torque pulsations depend on the relative duration of switching-on of the valves of the pulse converter \(\gamma\) and, at a constant commutation frequency during starting, have a maximum value at \(\gamma \cong 0.5\). Moreover, as the commutation frequency decreases, this maximum shifts toward smaller \(\gamma\) (Fig. 2). As the relative switching-on duration \(\gamma\) decreases, the magnitude of the motor electromagnetic torque decreases. This makes it possible to synthesize a specified dynamic characteristic under pulse control.
The investigations showed that, when controlling by changing the relative switching-on duration \(\gamma\), there is a noticeable reduction in the peaks of impact torques. In this case the motor torque can—
Fig. 1
Fig. 2
may be limited to a certain predetermined value, maintained unchanged throughout the entire starting time, or regulated in magnitude.
The possibility of controlling transient electromagnetic torques and of carrying out controlled starting of the motor with a specified dynamic characteristic expands the field of practical application of the pulse control systems under consideration.
The model was also used to investigate transient processes during application and removal of load, a step change in the control action, changes in the moment of inertia of the electric drive, and repeated switching-on from different initial speeds.
With an increase in the moment of inertia of the electric drive, the starting time increases; therefore the rate of change of the motor slip decreases, which creates conditions for a larger number of pulsations of the transient electromagnetic torque of the motor.
As the initial speed at the instant of switching on increases, the magnitude of the motor’s electromagnetic torque decreases.
The investigations carried out of electromechanical transient processes make it possible to conclude that the pulse system causes additional pulsations to appear in the curve of the motor’s electromagnetic torque, whose frequency is equal to the commutation frequency. The magnitude of these pulsations at a high commutation frequency is insignificant and does not exert a substantial influence on the nature of the transient process. Qualitatively, the transient processes in this case are analogous to transient processes under rheostatic speed control.
These circumstances make it possible to calculate electromechanical transient processes on an analog computer in the case \(\gamma=\mathrm{const}\) and \(f_k=\mathrm{const}\) without a model of the pulse system. For this purpose, in the system of differential equations one should take the equivalent value of the rotor-circuit resistance as
\[ r_2 = r_2' + R_d'(1-\gamma). \]
The main conclusions have been confirmed by experiments on an asynchronous motor of type MT-112-6. The discrepancy between the calculated and experimental values of the electromagnetic torque of the asynchronous motor is 10–15%.
Conclusions
-
Pulse control leads to the appearance, in the electromagnetic-torque curve of an asynchronous motor, of additional pulsations whose frequency is equal to the commutation frequency. As the commutation frequency increases, the additional torque pulsations decrease.
-
As the relative on-time decreases, the magnitude of the electromagnetic torque of the asynchronous motor decreases.
-
At an increased commutation frequency and unchanged \(\gamma\), the electromechanical transient processes are qualitatively analogous to those in an asynchronous electric drive with rheostatic control.
-
When \(\gamma\) is varied according to a specified law, controlled starting of an asynchronous motor with a prescribed dynamic characteristic is possible.
References
-
Yu. P. Kostyukov, O. G. Makarchenko. Speed control of an asynchronous motor by pulsed variation of the active resistance of the rotor circuit. In: Proceedings of the VIII Scientific and Technical Conference. Static Converters in Automation and Electric Drive. Tomsk, 1971.
-
O. G. Makarchenko. Development and Study of Asynchronous Electric Drives with Pulse Control of Rotational Speed. Author’s abstract of a Candidate dissertation. Tomsk, 1972.
-
Wu Zha Han. Study of the influence of saturation on the starting dynamics of an asynchronous motor. Elektrotekhnika, 1967, No. 8.