Calculation of the Control Angle in Continuous-Pulse Control Systems
Abstract Generated abstract
This paper derives a firing angle control law for continuous-pulse speed regulation of a DC motor supplied by an ionic rectifier. By equating the areas of current pulses through each valve under specified load limits, the authors obtain an expression for the firing angle as a function of relative voltage, permissible load, and the number of rectifier anodes. The resulting law is intended to keep pulse areas constant across the regulation range, simplifying motor heating calculations and bringing the mode closer to an optimal armature-current regime. An example for a PN-100 motor with a three-anode converter shows constant pulse area over the tabulated speed range, and the computed angle dependence can be approximated by two straight lines for practical implementation.
Full Text
Calculation of the Control Angle in Continuous-Pulse Regulation Systems
Yu. M. Achkasov, A. I. Zaitsev
(Presented by the scientific seminar of the Electromechanical Faculty)
In continuous-pulse regulation of the rotational speed of a DC motor in the URV-D system, depending on the depth of regulation, the firing angle of the ionic converter changes, while the character of the rectified voltage remains pulsed [1, 2]. The firing angle in this case is equal to the control angle and must be specified in advance for the load condition of interest to us.
Let us determine the law of variation of the firing angle \(\vartheta_z\) under continuous-pulse regulation as a function of the regulation range and the load on the motor shaft, that is, the dependence \(\vartheta_z = f(\varepsilon, t_{\mathrm{cp}})\), where \(\varepsilon\) is the relative value of the voltage determining the motor speed; \(t_{\mathrm{cp}}\) is the relative value of the average armature current of the motor.
To ensure the continuous-pulse regulation mode, the firing angle at speed \(n_1 < n\) must differ from the firing angle in the case of continuous regulation at the same speed \(n_1\). (It is meant that the load on the motor shaft is the same in magnitude in both cases.) This is clear from the fact that, in the continuous-pulse regulation mode, the rectifier operates with missed firings, while in the continuous regulation mode it operates without misses; therefore, to provide the very same load \((t_{\mathrm{cp}})\), the current pulses through each anode in the first mode must be larger in area than in the second mode. Consequently, the firing angle for the first (continuous-pulse) mode must be smaller than the firing angle of the second (continuous) mode.
To determine the dependence \(\vartheta_z = f(\varepsilon, t_{\mathrm{cp}})\), let us impose the condition: the regulation system in the continuous-pulse mode must ensure speed stabilization when the load varies within the limits:
\[ 0 < t_{\mathrm{cp}} < \lambda t_{\mathrm{n}}, \tag{1} \]
where \(\lambda\) is the coefficient of permissible motor current load \((\lambda = 1 \div 2.5)\);
\(t_{\mathrm{n}}\) is the relative value of the rated armature current of the motor.
In practice, speed stabilization is usually required under overloads not exceeding \(1.5 t_{\mathrm{n}}\).
Thus, with a load on the motor shaft equal to \(\lambda_n\), the system enters the continuous-regulation mode, thereby ensuring operation of the motor on the soft part of the mechanical characteristics (Fig. 1).
Fig. 1.
Diagram labels: vertical axis \(\varepsilon\); origin \(0\); horizontal axis \(t_{\mathrm{avg}}\); \(\lambda_n\); curves marked \(\vartheta_{31}\), \(\vartheta_{32} > \vartheta_{31}\), \(\vartheta_{33} > \vartheta_{32}\), \(\vartheta_{34} > \vartheta_{33}\), \(\vartheta_{35} > \vartheta_{34}\), \(\vartheta_{36} > \vartheta_{35}\).
With a load equal to \(\lambda_n\), a current \(i_{\mathrm{avg}\,m}\) flows through the motor armature, determined by the expression
\[ i_{\mathrm{avg}\,m}=\frac{mS_1}{2\pi}, \tag{2} \]
where \(m\) is the number of rectifier anodes;
\(S_1\) is the area of the current pulse passing through one valve [1].
\[ S_1=\left[\cos \vartheta_3-\cos\left(\vartheta_3+\frac{2\pi}{m}\right)-\varepsilon\frac{2\pi}{m}\right]. \tag{3} \]
On the basis of expressions (1) and (2), we can write
\[ i_{\mathrm{avg}\,m}=\frac{mS_1}{2\pi}=\lambda_n. \tag{4} \]
Substituting the value of \(S_1\) from expression (3), we obtain:
\[ i_{\mathrm{avg}\,m}=\lambda_n=\frac{m}{2\pi} \left[ \cos \vartheta_3-\cos\left(\vartheta_3+\frac{2\pi}{m}\right)-\varepsilon\frac{2\pi}{m} \right], \tag{5} \]
whence
\[ \cos \vartheta_3-\cos\left(\vartheta_3+\frac{2\pi}{m}\right) = \frac{2\pi}{m}\left(\varepsilon+\lambda_n\right). \tag{6} \]
Transform the left-hand side of expression (6) using the formula for the difference of two cosines:
\[ -2\sin \frac{\vartheta_3+\vartheta_3+\dfrac{2\pi}{m}}{2}\, \sin \frac{\vartheta_3-\vartheta_3-\dfrac{2\pi}{m}}{2} = \frac{2\pi}{m}(\varepsilon+\lambda_{\mathrm{н}}), \tag{7} \]
or
\[ \sin\left(\vartheta_3+\frac{\pi}{m}\right) = \frac{\pi(\varepsilon+\lambda_{\mathrm{н}})} {m\sin \dfrac{\pi}{m}} . \tag{8} \]
This trigonometric equation has an infinite set of roots, contained in two solutions:
\[ 1)\quad \vartheta_3=(2k+1)\pi-\frac{\pi}{m} -\arcsin\left[ \frac{\pi(\varepsilon+\lambda_{\mathrm{н}})} {m\sin \dfrac{\pi}{m}} \right]; \tag{9} \]
\[ 2)\quad \vartheta_3=2k\pi-\frac{\pi}{m} +\arcsin\left[ \frac{\pi(\varepsilon+\lambda_{\mathrm{н}})} {m\sin \dfrac{\pi}{m}} \right], \tag{10} \]
where \(k\) is any integer.
It is obvious that in the rectification mode the first solution will be valid, and since the firing angle is measured from the beginning of the sinusoid of the alternating voltage, expression (9) may be written in the following form:
\[ \vartheta_3=\pi-\frac{\pi}{m} -\arcsin\left[ \frac{\pi(\varepsilon+\lambda_{\mathrm{н}})} {m\sin \dfrac{\pi}{m}} \right]. \tag{11} \]
Thus, for a preliminarily selected motor we can determine the law of regulation of the firing angle \(\vartheta_3\), under which the areas of the current pulses in all regulation modes will be equal to one another. This brings the mode of continuous-pulse regulation closer to the optimal-control mode with respect to the motor armature current.
When the regulation law determined by expression (11) is observed, the calculations for determining the heating of the electric motor in the continuous-pulse regulation mode are substantially simplified. When the rectifier operates in the continuous-current region, the duration of the current pulses through each anode remains constant; consequently, the quadratic areas of the current pulses will also be equal to one another.
To check the selected electric motor for heating, it is necessary to determine the number of current pulses occurring during one period of the alternating voltage and the quadratic area of one current pulse passing through one valve [1].
The most laborious part of this calculation is determining the quadratic area of the current pulse; however, taking into account that it has to be determined only once, this method for checking the selected motor for heating may be considered quite suitable for engineering calculations.
In the example below, the control law for the firing angle \(\vartheta_з=f(\varepsilon)\) is calculated for a continuous-pulse control system for the speed of a PN-100 type motor: \(U_н=220\) V, \(I_н=58.7\) A, \(n_н=1470\) rpm, \(T_я=0.0272\) sec, \(U_м=311\) V.
The system must provide speed stabilization over the load range \(0<I<1.5 I_н\).
Table 1 gives the values of the firing angle, calculated from expression (11), for \(\lambda=1.5\), \(m=3\), and various values of \(\varepsilon\). The area of the current pulse through one valve was determined from expression (3).
Table 1
| \(\varepsilon\) | 0.71 | 0.7 | 0.6 | 0.5 | 0.4 | 0.3 | 0.2 | 0.1 | 0 |
|---|---|---|---|---|---|---|---|---|---|
| \(\vartheta_з^\circ\) | 30 | 38 | 60 | 72 | 81 | 90 | 97.5 | 105 | 112 |
| \(S_1\) | 0.24 | 0.24 | 0.24 | 0.24 | 0.24 | 0.24 | 0.24 | 0.24 | 0.24 |
As can be seen from the table, the area of the current pulse remains constant for all speed values; consequently, the quadratic area of the current pulse will also be constant.
Figure 2 shows the graphical dependence \(\vartheta_з=f(\varepsilon)\), constructed from Table 1. With sufficient accuracy this curve can be modeled by two straight lines, which substantially simplifies the practical implementation of this firing-angle control law [2].
Fig. 2.
Conclusions
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Continuous-pulse control of the rotational speed of an electric motor makes it possible, over the entire control range, to provide current pulses passing through one valve that are equal in area.
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These pulses do not change in magnitude when the load changes within permissible limits.
- The derived calculation formulas make it possible to determine the law of regulation of the ignition angle of an ionic converter, ensuring current pulses of equal area over the entire regulation range.
References
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Yu. M. Achkasov, V. A. Beinarovich, A. I. Zaitsev. Verification of motor heating in a medium-power ionic drive with pulse regulation of rotational speed. Izvestiya TPI, vol. 117, 1963.
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Yu. M. Achkasov, V. N. Mishin, A. A. Kuvshinov. A thyratron converter with continuous-pulse regulation of rectified voltage for low- and medium-power electric drives. Collection of PNTIPO No. 28-64-1266/53. Published by GOSINTI, 1964.