On the Influence of Electromagnetic Asymmetry of the Rotor on the Load Current of a Pulsed Machine-Commutated Generator
Abstract Generated abstract
This paper examines how electromagnetic asymmetry of the rotor affects the load current of a pulsed machine-commutated generator operating through a rectifier into an active load. Using a synchronous-generator model in rotating coordinates and simplifying the initial transient by treating the rotor circuits as superconducting, the authors derive expressions for the initial surge of the rectified load current as a function of direct- and quadrature-axis subtransient reactances and load resistance. The analysis shows that a symmetric rotor gives a minimum current condition, while increasing asymmetry, for example by raising or opening the quadrature-axis damper circuit, can increase the initial current pulse. Experimental oscillograms and further calculations are presented as support for the conclusion that rotor asymmetry may improve generator utilization and substantially increase pulse energy.
Full Text
Proceedings of the Tomsk Polytechnic Institute
of the Order of the October Revolution and the Order of the Red Banner of Labor
named after S. M. Kirov
Volume 284 1974
On the Influence of Electromagnetic Asymmetry of the Rotor on the Magnitude of the Load Current of a Pulsed Machine-Valve Generator
Yu. I. Ryabchikov, G. A. Sipailov, A. V. Loos
(Presented by the scientific seminar of the departments of electrical machines and apparatus and general electrical engineering)
The machine-valve generators currently used to obtain powerful unipolar current pulses of adjustable duration are, in most cases, a three-phase synchronous generator operating into a load through a rectifier assembled according to the Larionov bridge circuit.
In the course of experimental studies of the operating modes of generators of this type, as well as studies carried out on a mathematical model, it has been established that the current pulse of the generator in the case of an active load is influenced not only by the parameters of the generator along the direct axis and by the magnitude of the load resistance \(R_{\text{н}}\), but also, to a significant extent, by the parameters of the generator along the quadrature axis. In other words, the load current of a pulsed machine-valve generator depends on the electromagnetic asymmetry of the rotor, which, for a generator with non-salient poles, is determined by the parameters of the damper circuits along the direct and quadrature axes. In connection with the problems of optimal design, as well as the problems of increasing the power extracted from electric-machine pulsed sources, the study of this phenomenon is of quite definite interest. Below, in order to obtain expressions reflecting the quantitative and qualitative aspects of this phenomenon, analytical studies are given, based on the results of previous studies set forth in [1]. As shown in [1], in the case of an active load, the analysis of the electromagnetic processes of a pulsed machine-valve generator may be carried out using the equations of a synchronous three-phase generator operating into a symmetrical equivalent load \(R_{\text{э}}=\dfrac{2}{3}R_{\text{н}}\).
Written in rotating axes rigidly connected with the rotor, these equations, under known assumptions, have the form:
\[ \begin{aligned} p\Psi_d &= -\omega\Psi_q - Ri_d,\\ p\Psi_q &= \omega\Psi_d - Ri_q,\\ p\Psi_f &= u_f - r_f i_f,\\ p\Psi_{Dd} &= -r_{Dd}i_{Dd},\\ p\Psi_{Dq} &= -r_{Dq}i_{Dq}, \end{aligned} \tag{1} \]
\[ \begin{aligned} \Psi_d&=L_d i_d+L_{ad}i_f+L_{ad}i_{Dd},\\ \Psi_f&=L_{ad}i_d+L_f i_f+L_{ad}i_{Dd},\\ \Psi_{Dd}&=L_{ad}i_d+L_{ad}i_f+L_{Dd}i_{Dd},\\ \Psi_q&=L_q i_q+L_{aq}i_{Dq},\\ \Psi_{Dq}&=L_{aq}i_q+L_{Dq}i_{Dq}, \end{aligned} \]
where
\(p\) is the differentiation symbol; \(R=R_e+r\);
\(r\) is the active resistance of a generator phase; the remaining notations are conventional.
Since an exact solution of system (1) in general form does not appear possible, we shall confine ourselves to finding the initial values of the current surges, for which purpose we take the rotor circuits to be superconducting, i.e.,
\[ r_f=r_{Dd}=r_{Dq}=0,\qquad u_f=0. \tag{2} \]
If the connection of the generator to the load was preceded by no-load operation, then system (1) has the following initial conditions:
\[ \begin{aligned} \Psi_{d0}&=\Psi_{Dd0}=L_{ad}i_{f0},\\ \Psi_{f0}&=L_f i_{f0},\\ \Psi_{q0}&=\Psi_{Dq0}=0. \end{aligned} \tag{3} \]
Taking (2) and (3) into account, system (1) can be reduced to a system of two differential equations:
\[ \begin{aligned} L_d''p i_d+R i_d+\omega L_q'' i_q&=0,\\ L_q''p i_q+R i_q-\omega L_d'' i_d&=\omega L_{ad}i_{f0}. \end{aligned} \tag{4} \]
Its solution is:
\[ i_d=-\frac{p_2A}{p_1-p_2}\,e^{p_1t} -\frac{p_1A}{p_1-p_2}\,e^{p_2t}+A, \tag{5} \]
\[ i_q=\frac{p_2L_q''B+\omega L_{ad}i_{f0}}{(p_1-p_2)L_q''}\,e^{p_1t} -\frac{p_1L_q''B+\omega L_{ad}i_{f0}}{(p_1-p_2)L_q''}\,e^{p_2t}+B, \]
where
\[ A=-\frac{x_{ad}i_{f0}x_q''}{x_d''x_q''+R^2},\qquad B=-\frac{R}{x_q''}A, \]
\[ p_{1,2}=-\frac{R}{2}\left(\frac{1}{L_d''}+\frac{1}{L_q''}\right) \pm \sqrt{ \frac{R^2}{4}\left(\frac{1}{L_d''}-\frac{1}{L_q''}\right)^2-\omega^2 }. \tag{6} \]
Upon transformation to phase coordinates, the constant components in solution (5) give the initial value of the amplitude of the fundamental harmonic of the phase current:
\[ I_{mb}=\sqrt{A^2+B^2} = \frac{x_{ad}i_{f0}\sqrt{(x_q'')^2+R^2}}{x_d''x_q''+R^2} = \frac{E_{m0}\sqrt{(x_q'')^2+R^2}}{x_d''x_q''+R^2}. \tag{7} \]
The variable components of solution (5), in the general case, give a set of periodic and aperiodic components of the phase current. Investigations have shown that, for load values \(R_{\text{н}}\) close to the matched ones, the damping of these components occurs so rapidly that they have no appreciable effect on the magnitude of the load-current surge. This circumstance makes it possible, without allowing a large error, to neglect them, which substantially simplifies the subsequent analysis. Taking the foregoing into account, the expression for determining the initial surge of the rectified current may be represented in the form
\[ I_{\text{н}}=k_{\text{п}} I_{m\varphi}, \]
where
\[ k_{\text{п}} \text{ is a coefficient that takes into account the pulsations of the rectified current.} \]
Fig. 1a. Oscillogram of the load-current pulse with a symmetrical rotor
Fig. 1b. Oscillogram of the load-current pulse with the damper winding open along the transverse axis
Taking (7) into account, we finally write
\[ I_{\text{н}}=k_{\text{п}}\frac{E_{m0}\sqrt{(x_q^{\prime\prime})^2+R^2}}{x_d^{\prime\prime}x_q^{\prime\prime}+R^2}. \tag{8} \]
Expression (8) establishes the dependence of the initial current surge in the load on the parameters of the generator. It is not difficult, for example, to see that, in order to increase the current \(I_{\text{н}}\), it is necessary in designing such generators to take measures ensuring the minimum possible value of the subtransient inductive reactance along the longitudinal axis. Since it is difficult to establish directly from (8) the influence of the subtransient inductive reactance along the transverse axis, we differentiate (8) with respect to the parameter \(x_q^{\prime\prime}\). We obtain
\[ \frac{dI_{\text{н}}}{dx_q^{\prime\prime}} = \frac{k_{\text{п}}E_{m0}R^2\left(x_q^{\prime\prime}-x_d^{\prime\prime}\right)} {\left(x_d^{\prime\prime}x_q^{\prime\prime}+R^2\right)^2\sqrt{(x_q^{\prime\prime})^2+R^2}}. \tag{9} \]
Equating (9) to zero, we find that the point \(x_q''=x_d''\), for the current \(I_n\), as a function of the parameter \(x_q''\), is critical. Since, when passing through this point in the positive direction, the derivative changes sign from minus to plus, this point corresponds to a minimum determined by the expression:
\[ I_n = k_p \frac{E_{m0}}{\sqrt{(x_d'')^2 + R^2}} . \tag{10} \]
From this it may be concluded that, by introducing electromagnetic asymmetry of the rotor \((x_q'' \ne x_d'')\), a certain increase in the initial surge of the load current, determined by (8), can be achieved. In practice this can be accomplished by increasing the leakage inductive reactance of the damper winding along the transverse axis, or else by opening it (the case of limiting rotor asymmetry with salient poles). The latter is illustrated by experimental oscillograms of the load-current pulse shown in Fig. 1a and Fig. 1b. The oscillograms are made on the same scales; the no-load voltage, the magnitude of the active load, and the generator parameters along the longitudinal axis are constant. The rotational speed of the generator corresponds to a voltage frequency of 50 Hz; the magnetic circuit is not saturated. The decrease in load current during the pulse, compared with its initial surge, is explained by the gradual penetration of the armature-reaction field into the rotor circuits, owing to the fact that their active resistances are not zero.
The noted effect of current increase with rotor asymmetry can be used to improve the utilization factor of a pulsed machine-valve generator when operating into an active load. Thus, additional studies carried out on the basis of the relations derived above give grounds to believe that the pulse energy can be increased by a factor of 1.5–2 in comparison with the energy obtained from a generator with a symmetrical rotor.
References
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A. V. Loos, Yu. I. Ryabchikov, G. A. Sipailov. Investigation of transient processes of a valve pulsed generator. Izvestiya TPI, vol. 265, 1973.
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A. V. Loos, Yu. I. Ryabchikov. Mathematical modeling of a synchronous generator with rectifier load. Izvestiya TPI, vol. 242, 1972.