Electromagnetic Quasi-Steady-State Modes in Reversible Electric Drives with Series-Wound Motors under Pulse Control
Abstract Generated abstract
This paper analyzes electromagnetic quasi-steady-state processes in reversible DC electric drives with series-excited motors under pulse speed control, where separate diode shunting of the armature and field circuits can produce unequal average currents and ripple levels. Using idealized assumptions for motor parameters, supply, valves, and armature EMF, the authors derive piecewise analytical expressions for armature and field currents over a commutation period and formulate procedures for determining key quantities such as the transition interval, maximum current, minimum currents, and average currents. The proposed exact graphical-analytical or computational solution is supplemented by approximate formulas based on series expansion. Experimental comparison indicated discrepancies of up to 5 percent for the exact calculation method and up to 20 percent for the approximate method.
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Electromagnetic Quasi-Steady-State Conditions in Reversible Electric Drives with Series-Excited Motors under Pulse Control
A. I. Sapozhnikov, V. A. Beinarovich
(Presented by the scientific and technical seminar of the Research Institute of Automation and Electromechanics)
In reversible DC electric drives with series-excited motors, under pulse control of speed, the field-winding and armature circuits are separately shunted by diodes (Fig. 1,a). When the time constants of the armature and field circuits are unequal, the diagrams of the armature current \(i_{\mathrm{a}}\) and the field current \(i_{\mathrm{f}}\) have different forms (Fig. 2), which leads to inequality of the average values of these currents and of the magnitudes of their pulsations.
To make possible an analytical calculation of electromagnetic processes in quasi-steady-state conditions, the following assumptions are introduced:
the motor parameters are characterized by constant values of the resistances and inductances of the armature circuit \(r_{\mathrm{a}}, L_{\mathrm{a}}\) and the field circuit \(r_{\mathrm{f}}, L_{\mathrm{f}}\);
the EMF \(E\) in the armature winding does not change during the commutation period \(T\), owing to the constancy of the rotational speed and the damping of field-flux pulsations by the action of eddy currents;
the valves have ideal characteristics (the resistance changes discontinuously from zero to infinity);
the power supply is ideal (the internal resistance and inductance are equal to zero).
Under the indicated assumptions, the calculated equivalent circuit for asymmetric pulse control is shown in Fig. 1,b. Let us denote
\[ T_{\mathrm{a}}=\frac{L_{\mathrm{a}}}{r_{\mathrm{a}}}, \qquad T_{\mathrm{f}}=\frac{L_{\mathrm{f}}}{r_{\mathrm{f}}}, \qquad T_{\mathrm{e}}=\frac{L_{\mathrm{a}}+L_{\mathrm{f}}}{r_{\mathrm{a}}+r_{\mathrm{f}}} \]
as the electromagnetic time constants of the armature, field, and motor circuits;
\(I_{\max}\) as the maximum current value, the same for the armature and field circuits (Fig. 2);
\(I_{\mathrm{a,min}}, I_{\mathrm{f,min}}, I_{\mathrm{a,av}}, I_{\mathrm{f,av}}\) as the minimum and average current values of the armature and field (Fig. 2);
\[ \frac{T_{\mathrm{a}}}{T}=\frac{1}{a}, \qquad \frac{T_{\mathrm{f}}}{T}=\frac{1}{b}, \qquad \frac{T_{\mathrm{e}}}{T}=\frac{1}{\varepsilon}, \qquad \frac{t}{T}=\tau \]
as the time constants of the circuits and the current time in relative units.
With the adopted assumptions and notation, the solution of the differential equations of electrical equilibrium on the separate interva-
... of the commutation period, at a constant pulse duty factor \(\gamma\), give expressions for the instantaneous current values in the quasi-steady-state regime. Under the condition \(T_v>T_a\), the field current is determined (Fig. 2) on the interval \(0\le \tau \le 1-\gamma+\delta\) by
\[ i_v=I_{\max}e^{-b\tau} \tag{1} \]
and on the interval \(1-\gamma+\delta\le \tau \le 1\)
\[ i_v=\frac{U-E}{r_a+r_v} -\left(\frac{U-E}{r_a+r_v}-I_{v,\min}\right) e^{-(1-\gamma+\delta)\varepsilon}, \tag{2} \]
where
\[ I_{v,\min}=I_{\max}e^{-(1-\gamma+\delta)b}. \tag{3} \]
The armature current for \(T_v>T_a\) is determined (Fig. 2) on the interval \(0\le \tau \le 1-\gamma\).
Fig. 1
\[ i_a=\left(I_{\max}+\frac{E}{r_a}\right)e^{-a\tau}-\frac{E}{r_a}, \tag{4} \]
on the interval \(1-\gamma\le \tau \le 1-\gamma+\delta\)
\[ i_a=\frac{U-E}{r_a} -\frac{U}{r_a}e^{-(\tau-1+\gamma)a} +\left(I_{\max}+\frac{E}{r_a}\right)e^{-a\tau}, \tag{5} \]
and on the interval \(1-\gamma+\delta\le \tau \le 1\) the armature current is equal to the field current (Fig. 2) and is determined by equation (2).
The values of the interval \(\delta\) and of the current \(I_{\max}\) are calculated by a graphical-analytical method or on a computer by jointly solving two equations, the first of which is obtained by equating the values of the currents \(i_v\) from (1) and \(i_a\) from (5) at \(\tau=1-\gamma+\delta\), and the second from the equality of the currents \(i_v\) at the beginning of the commutation period from (1) at \(\tau=0\) and at the end from (2) at \(\tau=1\). To simplify the determination of \(\delta\) and \(I_{\max}\) under the same conditions, one may use approximate equations obtained from (1), (2), and (5) by expanding in a power series while retaining only its first two terms. In this case we obtain
\[ \delta= \frac{A[b(1-\gamma)+\varepsilon]+B\gamma} {ab-A(b-\varepsilon)+B}, \tag{6} \]
where
\[ A=\frac{Ea}{U-E}(1-\gamma); \qquad B=\frac{r_a\varepsilon}{r_a+r_v}(1-\gamma)(a-b); \]
\[ I_{\max}=\frac{U-E}{r_{\mathrm{я}}}\,\frac{a\delta-A}{(1-\gamma+\delta)(a-b)} . \tag{7} \]
Having determined \(\delta\) and \(I_{\max}\) for the specified values of \(\gamma\), \(E\), \(U\), and the known parameters, one can find the minimum values of the excitation current from (3) and of the armature current from (4) at \(t=1-\gamma\). The mean values of the currents are determined by integrating equations (1), (2), (4), (5):
\[ \begin{aligned} I_{\mathrm{я.\,av}}=I_{\max}\Bigg[ \frac{1-e^{-(1-\gamma+\delta)a}}{a} +\frac{1-e^{-(\gamma-\delta)\varepsilon}}{\varepsilon}\, e^{-(1-\gamma+\delta)b} \Bigg] &-\frac{E}{r_{\mathrm{я}}}(1-\gamma+\delta) +\frac{E}{a r_{\mathrm{я}}}\left(1-e^{-(1-\gamma+\delta)a}\right) +\frac{U}{r_{\mathrm{я}}}\delta -\frac{U}{a r_{\mathrm{я}}}\left(1-e^{-a\delta}\right) \\ &+\frac{U-E}{r_{\mathrm{я}}+r_{\mathrm{в}}}(\gamma-\delta) -\frac{U-E}{\varepsilon(r_{\mathrm{я}}+r_{\mathrm{в}})} \left(1-e^{-(\gamma-\delta)\varepsilon}\right), \end{aligned} \tag{8} \]
\[ I_{\mathrm{в\,av}} =I_{\max}\left\{ \frac{1}{b} +e^{-(1-\gamma+\delta)b} \left[ \frac{1}{\varepsilon}\left(1-e^{-(\gamma-\delta)\varepsilon}\right) -\frac{1}{b} \right]\right\} +\frac{U-E}{r_{\mathrm{я}}+r_{\mathrm{в}}} \left[ \gamma-\delta -\frac{1}{\varepsilon}\left(1-e^{-(\gamma-\delta)\varepsilon}\right) \right]. \tag{9} \]
Fig. 2
Experimental checks have shown that the calculation method presented gives a discrepancy with experiment of up to \(5\%\) when the exact solution is used, and up to \(20\%\) for the approximate one.