DETERMINATION OF THE CHARGING EFFICIENCY OF A CAPACITIVE STORAGE DEVICE BY MATHEMATICAL MODELING

V. A. Kochegurov, A. A. Tereshchenko

(Presented by the scientific seminar of the UVL)

In the development of pulsed current sources with capacitive energy storage devices, it is often necessary to evaluate their energy indicators and, first of all, their efficiency. This is especially true of powerful current sources, where low efficiency leads to excessive consumption of active energy from the network.

The magnitude of the efficiency and the method of its analytical evaluation are determined by the type of charging device. In practical circuits, charging of a capacitive storage device is carried out through active resistances and inductances. In this case, the resistances and inductances may be connected both before the rectifier and after it [1].

Let us consider the analytical calculation of efficiency for the case of charging capacitors from a three-phase bridge rectifier through an active resistance at a constant control angle. We shall construct a single-loop equivalent circuit (Fig. 1,b). The voltage applied to the terminals of the equivalent circuit is determined from the condition

\[ U_{\mathrm{в}}=\sqrt{3}U_m\cos\alpha, \tag{1} \]

where \(U_m\) is the amplitude of the phase emf;
\(\alpha\) is the rectifier control angle.

The differential equation describing the operation of the circuit (Fig. 1,b) has the form:

\[ U_{\mathrm{в}}=U_c+2\Delta U_{\mathrm{в}}+R_{\mathrm{э}}i+L_{\mathrm{э}}\frac{di}{dt}, \tag{2} \]

where \(U_c\) is the voltage across the capacitance;
\(\Delta U_{\mathrm{в}}\) is the voltage drop in the valve arc;
\(R_{\mathrm{э}}, L_{\mathrm{э}}\) are the equivalent active resistance and inductance of the equivalent circuit;
\(i\) is the current in the loop.

The solution of equation (2) in relative units has the form:

\[ i=\left[\cos\Theta\cdot\sin(\alpha-\Theta+\omega t)-\varepsilon\right] +\left[\varepsilon-\cos\Theta\sin(\alpha-\Theta)\right]e^{-\omega t\operatorname{ctg}\Theta}, \tag{3} \]

Fig. 1.
a — circuit for charging a capacitive storage device; b — equivalent circuit of a capacitive generator; c — electronic model for determining the efficiency of capacitor charging.

where

\[ \varepsilon=\frac{U_c+2\Delta U_{\mathrm{v}}}{U_m}; \qquad i=\frac{i_{\mathrm{n}}(R_3+R_{\mathrm{e}})}{U_m}; \qquad \operatorname{tg}\Theta=\frac{1.5\,\omega L}{R_3+1.75R_{\mathrm{e}}}, \]

\(\Theta\) is the phase angle of the total resistance of the equivalent circuit. The expressions given make it possible to calculate numerically or graphically [1] the charging current as a function of \(\operatorname{tg}\Theta\), \(\alpha\), \(\varepsilon\), \(\lambda\), and to determine the voltage across the capacitance at a given instant of time:

\[ U_c(t)=U_{c0}+\frac{1}{C}\int_0^\lambda i_{\mathrm{n}}(t)\,dt, \tag{4} \]

where \(U_{c0}\) is the residual voltage on the capacitors. Then the weighted-average value of the efficiency over a cycle for the case under consideration is written as

\[ \eta= \frac{ \frac{1}{T}\int_0^{t_3}\left[\left|\sqrt{3}\,U_m\cos\alpha(t)-\Delta U(t)-U_{c0}(t)\right|\cdot i_{\mathrm{n}}(t)\,dt -\int_0^{t_3}U_{c0}(t)\cdot i_{\mathrm{n}}(t)\,dt\right] }{ n_{\mathrm{v}}\Delta U_{\mathrm{v}} I_{\mathrm{n}}\cdot\frac{1}{m} +\sum_{m=A,B,C} I_{\mathrm{cp}}^{2}\cdot r +I_{\mathrm{n}}^{2}R_3 +\Delta P_{\mathrm{u}} +\Delta P_{\mathrm{sn}} }, \tag{5} \]

where \(T\) is the repetition period of the charging pulses,
\(t_3\) is the capacitor recharging time,
\(\Delta U(t)\) is the voltage loss across the charging resistances, valves, and transformer,
\(i_{\mathrm{n}}(t)\) is the charging current,
\(n_{\mathrm{v}}\) is the number of series-connected valves,
\(I_{\mathrm{cp}}\) is the average current of the rectifier valve,
\(m\) is the number of rectifier phases,
\(r\) is the resistance of a transformer phase,
\(I_{\mathrm{n}}\) is the average rectified current,
\(R_3\) is the charging resistance,
\(\Delta P_{\mathrm{u}}\) is the power consumed by the converter control system,
\(\Delta P_{\mathrm{sn}}\) is the power consumed by the auxiliary-services system (filament circuits, cooling system, etc.).

The expression obtained makes it possible to calculate analytically the charging efficiency, but it requires a large amount of computational work, since the charging current and the voltage across the capacitance do not remain constant during the charging process. The most advanced engineering method for studying such devices is the use of analog computers.

In Fig. 1, в, an electronic model is presented that makes it possible to determine the efficiency when charging capacitors through \(LR\) connected into the charging circuit.

The electronic model is compiled from the equations describing the operation of the circuit [2, 3] and from equation (5), which gives the dependence of the efficiency on the elements and parameters of the circuit. Operational amplifiers 1, 2 with tube diodes in the feedback circuit simulate the controlled valve.

The voltage at the output of the divider unit is proportional to the efficiency per cycle.

The method considered for analyzing the energy characteristics of pulse generators with the aid of an analog computer can find broad practical application.

REFERENCES

1. A. A. Bulgakov. Fundamentals of the Dynamics of Controlled Rectifier Systems. Publishing House of the Academy of Sciences of the USSR, 1963.

2. V. A. Kochegurov, A. A. Tereshchenko. Investigation on an analog computer of a closed-loop automatic-control system containing an ionic converter. Izv. TPI, vol. 162, 1967.

3. V. A. Kochegurov, A. A. Tereshchenko. Application of analog computers to the analysis of electromagnetic processes in three-phase rectifiers. Izv. TPI, vol. 159, 1968.