Reactive Converters of Current Sources into Voltage Sources
Abstract Generated abstract
This paper analyzes reactive inductive-capacitive circuits that are dual to Boucherot converters, intended to transform a current source into a voltage source. Using phasor relations and vector diagrams, the authors derive expressions for two single-phase converter variants and show that, under the ideal tuning condition of equal inductive and capacitive reactances, the load voltage remains independent of load resistance while the load current changes accordingly. The analysis is extended to nonideal circuits with active losses, where voltage constancy is only approximate and depends on the relation between load impedance and branch resistances. The paper also notes possible applications in charging inductive storage elements and suggests broader families of such converters, including T-shaped, bridge, coupled, and three-phase forms.
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REACTIVE CONVERTERS OF CURRENT SOURCES INTO VOLTAGE SOURCES
S. P. Kononov, A. S. Belokopytov
(Submitted by the scientific seminar of the Departments of Electrical Machines
and General Electrical Engineering)
Reactive (inductive-capacitive) converters of voltage sources into current sources (Boucherot circuits) are described quite well in the literature. Many articles \([1, 2]\), etc., and even entire monographs \([3]\) are devoted to them. This is explained by the promise of such circuits in pulsed electric power engineering as intermediate devices when charging capacitive storage units from an AC-voltage network.
In accordance with the principle of duality, it may be asserted that there exist circuits dual to Boucherot circuits, which can also be used in engineering. The present article is devoted to the analysis of such circuits, which make it possible to convert a current source into a voltage source. In the literature known to the authors there is only mention of similar circuits, but no analysis of them.
The processes in the first of the circuits (Fig. 1), connected to a current source \(\dot{i}\), are described by the following expressions:
Fig. 1
The output voltage is
\[ \dot{U}=\dot{i}\,\frac{j x_L\left(z_H-jx_C\right)}{z_H+j\left(x_L-x_C\right)}, \]
the inductor current is
\[ \dot{i}_L=\dot{i}\,\frac{z_H-jx_C}{z_H+j\left(x_L-x_C\right)}, \]
the load current is
\[ \dot{i}_H=\dot{i}\,\frac{j x_L}{z_H+j\left(x_L-x_C\right)}, \]
the voltage across the capacitance is
\[ \dot U_C=\dot I\,\frac{x_L x_C}{z_n+j(x_L-x_C)}, \]
the voltage across the load is
\[ \dot U_n=\dot I\,\frac{x_L z_n}{z_n+j(x_L-x_C)}. \]
If \(x_L=x_C=x\), then we have
\[ \dot U=\dot I\left(jx+\frac{x^2}{z_n}\right),\qquad \dot I_L=\dot I\left(1-j\frac{x}{z_n}\right),\qquad \dot I_n=\dot I j\frac{x}{z_n}, \]
\[ \dot U=\dot I\,\frac{x^2}{z_n},\qquad \dot U_n=\dot I jx . \]
It is evident that when \(x_L=x_C\), the load voltage does not depend on its resistance. This is explained by an automatic increase in the current when the load resistance decreases. Analysis of the vector diagram (Fig. 2)
Fig. 2
for the case when \(x_L=x_C=x\), for three different load resistances \(z_n\), confirms the considerations expressed above. It is seen that the load voltage, which leads the current \(I\) of the source by \(\pi/2\), is unchanged. A change in the resistance \(z_n\) affects only the load current and the voltage across the other elements of the circuit. The apparent power of the load \(S=\dot U_n \dot I_n\) is equal to the active power and is determined both by the load resistance and by the source current.
Such a circuit may find practical application, for example, for charging inductive storage elements \(L_n\) connected in place of the resistance \(z_n\) (Fig. 3). The advantage of the circuit is the possibility of obtaining, in the inductance, a current considerably exceeding the source current. Theo-
Fig. 3
of \(z_n\) (Fig. 3). The advantage of the circuit is the possibility of obtaining, in the inductance, a current considerably exceeding the source current. Theo-
Fig. 4
theoretically, in the presence of ideal elements, the charging current can increase to an infinitely large value even when supplied from a source of insignificant current.
In the real circuit (Fig. 4), the energy losses in the inductance and capacitance of the circuit cannot be neglected. In this case we have
\[ \dot U = \dot i\, \frac{ r_L z_H + r_C r_L + x_C x_L + j\left(x_L r_C - x_C r_L + z_H x_L\right) }{ r_L + r_C + z_H + j\left(x_L - x_C\right) }, \]
\[ \dot i_L = \dot i\, \frac{ z_H + r_C - jx_C }{ r_L + r_C + z_H + j\left(x_L - x_C\right) }, \qquad \dot i_H = \dot i\, \frac{ r_L + jx_L }{ r_L + r_C + z_H + j\left(x_L - x_C\right) }, \]
\[ \dot U_C = \dot i\, \frac{ x_L x_C - jr_L x_C }{ r_L + r_C + z_H + j\left(x_L - x_C\right) }, \qquad \dot U_H = \dot i z_H\, \frac{ r_L + jx_L }{ r_L + r_C + z_H + j\left(x_L - x_C\right) }. \]
If \(x_L = x_C = x\), then
\[ \dot U = \dot i\, \frac{ r_L z_H + r_C r_L + x^2 + j\left(r_C - r_L + z_H\right) }{ r_C + r_L + z_H }, \]
\[ \dot i_H = \dot i\, \frac{ r_L + jx }{ r_L + r_C + z_H }, \qquad \dot U_H = \dot i z_H\, \frac{ r_L + jx }{ r_L + r_C + z_H }. \]
It is evident that in such a nonideal case the voltage across the load already depends on its resistance \(z_H\), increasing as the latter grows. Only when \(r_L + r_C \ll |z_H|\) is the voltage across the load practically constant, while the current varies inversely proportionally to the load resistance.
There is also another variety of such a single-phase reactive converter (Fig. 5), in which the load resistance is connected in series with the inductance of the parallel circuit. In this case, an unchanged voltage is likewise maintained at the terminals of \(z_H\), if it is connected to the current source \(\dot i\). In this case we have
Fig. 5
\[ \dot U=-\dot I j x_C \frac{z_H+j x_L}{z_H+j(x_L-x_C)}, \qquad \dot i_C=\dot I \frac{z_H+j x_L}{z_H+j(x_L-x_L)}, \]
\[ \dot i_H=-\dot I j x_C \frac{1}{z_H+j(x_L-x_C)}, \qquad \dot U_L=\dot I \frac{x_L x_C}{z_H+j(x_L-x_C)}. \]
\[ \dot U_H=-\dot I j \frac{x_C z_H}{z_H+j(x_L-x_C)}. \]
If \(x_L=x_C=x\), then we find that
\[ \dot U=\dot I\left(\frac{x^2}{z_H}-jx\right), \qquad \dot i_C=\dot I\left(1+j\frac{x_C}{z_H}\right), \qquad \dot i_H=-\dot I \frac{jx}{z_H}, \]
\[ \dot U_L=\dot I \frac{x^2}{z_H}, \qquad \dot U_H=-\dot I jx. \]
The nature of the process in the system is conveniently evaluated by using the vector diagram (Fig. 6), constructed for three different resistances \(z_H\). In this case the inductive storage unit \(L_H\) of the pulsed electroenergetic system is also connected in place of the resistance \(z_H\) (Fig. 7). As in the preceding circuit, the current of such a storage unit can be considerably greater than the source current, which is a positive property of the system.
Fig. 6
Fig. 7
Fig. 8
In a real circuit (Fig. 8), when the active resistances of the inductive and capacitive branches cannot be neglected, the processes are described by the relations
\[ \dot U=\dot I \frac{r_C z_H+r_L r_C+x_L x_C+j(x_L r_C-r_L x_C-z_H x_C)} {r_C+r_L+z_H+j(x_L-x_C)}, \]
\[ \dot i_C=\dot I \frac{r_L+jx_L+z_H}{r_C+r_L+z_H+j(x_L-x_C)}, \qquad \dot i_H=\dot I \frac{r_C-jx_C}{r_C+r_L+z_H+j(r_L-x_C)}. \]
\[ \dot{U}_L=\dot{I}_j x_L \frac{r_C-jx_C}{r_C+r_L+z_H+j(x_L-x_C)},\qquad \dot{U}_H=\dot{I}z_H \frac{r_C-jx_C}{r_C+r_L+z_H+j(x_L-x_C)}. \]
If \(x_L=x_C=x\), then we have
\[ \dot{U}=\dot{I} \frac{r_Cz_H+r_Lr_C+x^2+jx(r_C-r_L-z_H)}{r_C+r_L+z_H}, \]
\[ \dot{I}_H=\dot{I}\frac{r_C+jx}{r_C+r_L+z_H},\qquad \dot{U}_H=\dot{I}z_H\frac{r_C-jx_C}{r_C+r_L+z_H}. \]
It is evident that in the real circuit the voltage across the load is no longer constant. It depends on the resistance \(z_H\), and only when \(|z_H|\gg r_L-r_C\) is the voltage across the load close to the ideal.
Analysis shows that, in addition to those described, a large number of other varieties of reactive converters of current sources into voltage sources can be created: single-phase T-shaped and bridge converters, with inductive coupling and without it, and three-phase converters.
REFERENCES
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L. C. Green, I. B. Kuper. Constant Current Sources, Rev. Scient. Instruments, New-York, Vol. 11, 1940.
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H. K. Lennings. Charging Large Capacitor Bank in Thermonuclear Research, Electric Eng., No. 6, 1961.
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A. N. Milyakh, B. E. Kubyshkin, I. V. Volkov. Inductive-Capacitive Converters. Kiev, 1967.