ANALYSIS OF THE POWER FACTOR OF CHARGING CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH AN INCREASED REPETITION RATE WHEN POWERED FROM A RECTIFIED-VOLTAGE SOURCE

Yu. B. Fortes, V. N. Safronov

(Presented by the scientific seminar of the High-Voltage Research Institute.)

The wide use of pulsed voltages and currents in applied circuits has led, over the last 10–15 years, to the development of new methods for generating periodic pulses, to the creation of new and the improvement of existing types, circuits, and designs of pulse generators. An ever broader range of technological processes is being covered by the application of pulsed-current generators (PCG), high-voltage pulse generators (HVPG), Goryev generators, etc. Thus, pulsed-current generators have found application in such technological processes as pulse welding [1], electroerosion machining of metals [2], in a number of processes involving the electrohydraulic effect [3], and so on. High-voltage pulse generators are also undergoing changes in their application. At the initial stage of their development, HVPGs were used only as simulators of atmospheric-electricity discharges for pulse testing and investigations of the insulation of high-voltage electrical equipment. In this case, the problems of pulse formation and transmission were brought to the fore, while less attention was paid to the energy aspect of pulse installations. Subsequently, the field of application of HVPGs continuously expanded, simultaneously with the improvement of charging circuits and of the structural design of generators, and with the simultaneous solution of a number of technical requirements imposed on HVPGs. At present, high-voltage pulse generators have found wide application in physics laboratories in studies of the structure of the nucleus, in industrial installations for accelerating technological processes, and in the mining industry in drilling and crushing rocks [4]. Recently, high-voltage pulse generators with an increased repetition rate, providing a pulse front \(\tau_f = 10^{-6} \div 10^{-7}\) sec, an amplitude of 180–300 kV, and an energy per pulse of 200–3000 J, have become widespread in engineering. These generators are used in the study of the physics of breakdown of solids, in investigations of the multi-pulse strength of liquid and solid insulation, and also find broad application as technological installations for the destruction of solids.

The use of high-voltage pulse generators with an increased repetition rate in technological installations, along with the usual requirements [4], imposes additional re-

requirements of high reliability and stability of operation in very long-duration modes, high efficiency, and power factor of the charging circuits.

In the literature there are no systematic data on the evaluation of the efficiency and power factor of the charging circuits of high-voltage pulse generators with an increased repetition frequency.

In the present article an analysis is made of the power factor of the charging circuits of frequency pulse voltage generators operating in the “self-starting” mode when charged from a rectified-voltage source.

With a number of accepted assumptions [5], the equivalent circuit of the charging circuit of a frequency pulse generator can be represented by an RLC oscillatory circuit, and it may be assumed that the equivalent resistance \(R\) includes the resistance of the charging transformer, the ohmic resistance of the protective choke, and the resistance of the rectifier unit; the equivalent inductance \(L\) includes the inductance of the charging transformer and the inductance of the protective choke. The initial data for calculating the charging circuits of frequency generators are: the pulse amplitude, the repetition frequency \(f\), the value of the charging capacitance \(C\), or the energy of the output pulse. The task of the analysis is to determine the optimal ratios \(\frac{\omega_0}{\omega}\) and the parameters of the RLC circuit that ensure the maximum power factor in the nominal operating mode.

In the course of the analysis, calculations were carried out for the charging modes of a capacitor bank at various specified circuit parameters and generator repetition frequencies.

The following assumptions were adopted:

1. The parameters of the charging circuit of frequency generators are linear.

2. The discharge time of the generator is incommensurably small compared with the charging period of the capacitor bank, and the high-voltage pulse generator fires immediately upon reaching \(U_c=\mathrm{const}\).

3. The firing angle \(\psi_i\) is taken equal to 0, since there is a whole series of modes in which the generator necessarily arrives at \(\psi_i=0\).

In carrying out the calculations, the values of the charging capacitance of the high-voltage pulse generator were taken in the range \(0.1\)—\(1\ \mu\mathrm{F}\). The repetition frequency of the generators was from \(2.5\ \mathrm{s}^{-1}\) to \(50\ \mathrm{s}^{-1}\), and the value of the active resistance of the discharge circuit was \(0\)—\(10000\ \Omega\).

The indicated quantities cover the range of variation of the parameters and operating modes of pulse generators used in electro-pulse technology.

The power factor of the charging circuit \(k_{\mathrm{M}}\) is the ratio of the active power consumed by the load to the input volt-amperes.

\[ k_{\mathrm{M}}=\frac{P}{I_g\cdot \dfrac{U_m}{\sqrt{2}}} = \frac{\dfrac{C U_c^2}{2}\cdot f}{I_g\cdot \dfrac{U_m}{\sqrt{2}}} \tag{1} \]

Introducing a number of coefficients, formula (1) can be represented in the following form, convenient for analysis:

\[ k_{\mathrm{M}}=\frac{k^2}{k_1}\cdot \frac{C\cdot f}{\sqrt{2}}, \tag{2} \]

where

\[ k_1=\sqrt{\frac{1}{T_n}\sum_{n=1}^{n=\infty}\int_{0}^{t_3}\varphi^2(t)\,dt}. \]

In turn,

\[ \varphi(t)=\frac{1}{z}\left\{\sin(\omega t+\psi_i)-e^{-bt}\left[\frac{\omega_0^2}{\omega\omega'}\cdot\cos\psi_i-\frac{b}{\omega'}\sin\psi_i\times\sin\omega't+\right.\right. \]

\[ \left.\left.+\sin\psi_i\cos\omega't\right]\right\}, \]

\[ k=\frac{U_c}{U_m}=\frac{1}{z\omega C}\left\{\sin(\omega t+\psi_u)-e^{-bt}\left[\left(-\frac{b}{\omega_0}\sin\psi_u+\right.\right.\right. \]

\[ \left.\left.\left.+\frac{\omega}{\omega_0}\cos\psi_u\right)\sin\omega't+\sin\psi_u\cdot\cos\omega't\right]\right\}. \]

Here \(U_c\) is the generator firing voltage;
\(U_m\) is the amplitude of the charging voltage;
\(T_n\) is the pulse repetition period;
\(C\) is the charging capacitance of the capacitor bank;
\(n\) is the number of half-periods of the rectified voltage of one sign (in our case \(n\) varied from 1 to 20, which corresponded to a change in the generator firing frequency from 50 to 2.5 pulses/sec);
\(\omega_0\) is the natural angular frequency of the oscillatory circuit;
\(\omega\) is the angular frequency of the mains, equal to 314;
\(\psi_i\) is the current phase;
\(\psi_u\) is the voltage switching angle.

From formula (2) it is evident that the power factor does not depend on the absolute values of the charging voltage \(U_m\) and the voltage across the capacitance \(U_c\), but is a function of their ratio. This circumstance makes it possible to analyze the power factor independently of whether \(U_c\) or \(U_m\) is taken as the constant quantity.

The coefficients \(k\) and \(k_1\) in formula (2) are determined by the parameters of the charging circuit and depend on the generator firing frequency. Table 1 gives the values of \(k\) and \(k_1\) when the firing frequency changes from 50 to 2.5 Hz for one of the variants considered in the analysis.

Table 1

Fig. 1. Dependence of the power coefficient on the ratio \(\omega_0/\omega\) for various values of the charging-circuit resistance. The charging capacitance is \(1 \times 10^{-6}\) F, switching angle \(\Psi_u = 0\).
1, 2, 3 — generator firing frequency 25 pulses/sec.
4, 5, 6 — generator firing frequency 10 pulses/sec.

In Figs. 1 and 2 typical calculated dependences are presented of the power coefficient of the charging circuit of generators on the ratio \(\omega_0/\omega\), for a constant firing frequency and for various values of the resistance \(R\) and capacitance \(C\) of the circuit, respectively. From the dependence shown in Fig. 1 it is seen that the power coefficient decreases with increasing values of the equivalent resistance over the entire range of variation of \(\omega_0/\omega\).

In addition, the general form of the dependences \(k_m = \varphi(\omega_0/\omega)\), both in Fig. 1 and in Fig. 2, is not identical at different firing frequencies and equal other conditions.

If, at a generator firing frequency of 25 pulses/sec, the dependence \(k_m = \varphi(\omega_0/\omega)\), curve 2 in Fig. 2, has a maximum at the value \(\omega_0/\omega = 0.3\), then at a generator firing frequency of 10 pulses/sec (curve 3 in Fig. 2) the dependence \(k_m = \varphi(\omega_0/\omega)\) has no maximum; the values of \(k_m\) decrease with increasing ratio \(\omega_0/\omega\).

The variation of the power coefficient with the generator firing frequency at constant circuit parameters is represented by the typical dependences in Figs. 3 and 4. From Fig. 3 it is seen that, with increasing values

Fig. 2. Dependence of the power coefficient on the ratio \(\omega_0/\omega\) for different values of the discharge capacitance. Resistance of the charging circuit \(R=5000\ \Omega\), firing angle \(\Psi_u=0\).

1 — for \(C=0.625\times10^{-6}\ \mathrm{F}\) at a firing frequency of \(25\) pulses/s,
2 — for \(C=1\times10^{-6}\ \mathrm{F}\) at a firing frequency of \(25\) pulses/s,
3 — for \(C=1\times10^{-6}\ \mathrm{F}\) at a firing frequency of \(10\) pulses/s,
4 — for \(C=0.625\times10^{-6}\ \mathrm{F}\) at a firing frequency of \(10\) pulses/s.

resistances, the power coefficient of the charging circuits decreases throughout the entire range of increase in the generator firing frequencies from 0 to \(50\) pulses/s.

The nature of the dependence of the power coefficient on the generator firing frequency is determined by the ratio of angular frequencies \(\omega_0/\omega\), all other conditions being equal. At \(\omega_0/\omega=0.2;\ 0.3\) (Fig. 4), as the generator firing frequency increases from 1 to \(50\) pulses/s, the power coefficient rises to its maximum value and then decreases; at \(\omega_0/\omega=0.4;\ 0.5\), with increasing generator firing frequency, the power coefficient increases without the presence of an optimum in this frequency interval.

It is also evident from Fig. 4 that in the region of low generator firing frequencies \((1—10\) pulses/s) higher values of the power coefficient occur at smaller ratios \(\omega_0/\omega\), whereas in the region of higher firing frequencies \((f>20\ \frac{\text{pulses}}{\text{s}})\) the generator has higher values of the power coefficient ...

Fig. 3. Variation of the power coefficient with the generator triggering frequency at various values of the charging-circuit resistance. Charging capacitance \(C = 1 \times 10^{-6}\ \mathrm{F}\), \(\omega_0/\omega = 0.3\); \(\Psi_{\mathrm{u}} = 0\).

larger ratios \(\omega_0/\omega\) correspond to \(K_m\). In addition, in the range of low generator triggering frequencies, a given triggering frequency corresponds to a single optimal ratio \(\omega_0/\omega\), which ensures the maximum value of the power coefficient for the given initial parameters; whereas in the range of high frequencies, the larger the ratio \(\omega_0/\omega\), the higher the value of the power coefficient, all other conditions being equal.

Conclusions

1. When the GIN operates in the “self-triggering” mode, the power coefficient of the GIN charging circuits is determined by the values of the ohmic resistance in the circuit, by the ratio of angular frequencies \(\omega_0/\omega\), and by the generator triggering frequency.

2. When supplied from a rectified-voltage source, the maximum values of the power coefficient are obtained at \(R = 0\), which is unrealistic for practical circuits, and even in this ideal case does not exceed 0.45.

Fig. 4. Dependence of the power coefficient of the charging circuit on the generator triggering frequency for various ratios \(\omega_0/\omega\). Charging capacitance
\(C = 0.625 \times 10^{-6}\ \mathrm{F}\), \(R = 5000\ \Omega\), \(\Psi_{\mathrm{i}} = 0\)

REFERENCES

1. A. L. Livshits, I. S. Rogachev. Generators of periodic high-current pulses. Gosenergoizdat, 1959.
2. A. L. Livshits. Electroerosion machining of metals. Mashgiz, 1957.
3. L. A. Yutkin. The electrohydraulic effect. Mashgiz, 1955.
4. S. M. Smirnov, P. V. Terent'ev. High-voltage pulse generators. Moscow, Energiya, 1964.
5. I. I. Kalyatskii, R. E. Klein, V. S. Barantsev, P.T.E. 2.108, 1964.