ON THE SPECIFIC FEATURES OF USING PULSED BREAKDOWN AT ULTRA-HIGH VOLTAGES

A. F. Usov

(Presented by the scientific seminar of the Department of High-Voltage Engineering)

In recent times, in various fields of technology, the use of pulsed-voltage generators has found wide application for generating shock waves in liquid media and solid dielectrics for the purposes of destroying solid bodies, studying the kinetics of chemical reactions that usually proceed at high pressures and temperatures, stamping, processing forgings, and other technological operations [1—6].

According to most published works, the operating voltages of the generators amount to tens of kilovolts, while the generator capacitance “in the pulse” is units and even tens of microfarads. The need for this is caused by the desire to increase the energy stored in the capacitors, which determines the power developed in the spark channel. Some authors [2, 7], incorrectly interpreting the fact that the developed power increases with an increase in the interelectrode gap, mistakenly believe that aperiodization of the process contributes to an increase in the power in the discharge. It must also be taken into account that increasing the generator capacitance “in the pulse” leads to an increase in the oscillation period, that is, the rate of rise of the power decreases. Even in a single-capacitor device, with the inductance of the discharge circuit reduced to a minimum, the rise time of the power amounted to several microseconds [7].

A considerable increase in the power in the discharge and in the rate of its rise can be obtained by using pulsed-voltage generators for hundreds of kilovolts with a generator capacitance “in the pulse” of several tens of thousands of picofarads.

The possibility of using high operating voltages is due to the fact that, for voltage-pulse action times of \(10^{-7}\) sec, the breakdown strength even for extremely nonuniform fields amounts to hundreds of kV/cm [8]. In uniform fields, the breakdown strength for water at such action times, according to [9], may reach even 2 MV/cm.

However, when assessing the possibility of using generators with an operating voltage of hundreds of kilovolts, it is necessary to take into account that the energy losses before breakdown are proportional to the square of the operating voltage, and generators in a multistage design have considerable inductances in the discharge circuit. Therefore it was of interest to estimate, taking into account the above, the efficiency of using

such generators and to consider the possibility of improving them so as to adapt them to the specific conditions of operation into a low-resistance load.

In analyzing the discharge circuit of the generator, we made the assumption that the resistance of the spark channel remains constant over the entire range of the time investigated. In doing so, we proceeded from the results of estimates of the spark-channel resistance as a function of time for liquid and solid dielectrics, obtained under similar operating conditions of the generator [10, 2]. In the breakdown of water the resistance of the spark channel is 4–10 ohms; in the breakdown of solid dielectrics, several tens of ohms.

The maximum value of the current in the \(R-L-C\) circuit may be written as

\[ I_{\max}=U\sqrt{\frac{C}{L}}\,e^{-\frac{b\varepsilon}{\omega}}, \tag{1} \]

where

\[ \varepsilon=\operatorname{arctg}\frac{\omega}{b},\qquad b=\frac{R}{2L},\qquad \omega=\sqrt{\frac{1}{LC}-\frac{R^{2}}{4L^{2}}}. \]

Introducing the notation

\[ \beta=\frac{C}{C_{\mathrm{cr}}}=\frac{C}{4L/R^{2}}, \]

we have

\[ I_m=U\sqrt{\frac{C}{L}}\exp\left[ -\frac{\operatorname{tg}^{-1}\sqrt{\frac{1}{\beta}-1}} {\sqrt{\frac{1}{\beta}-1}} \right], \tag{2} \]

\[ P_m=\frac{2W_0R}{L}\exp\left[ -\frac{\operatorname{tg}^{-1}\sqrt{\frac{1}{\beta}-1}} {\sqrt{\frac{1}{\beta}-1}} \right], \tag{3} \]

where \(C_{\mathrm{cr}}\) is the generator capacitance corresponding to the aperiodicity boundary, and \(W_0\) is the energy stored in the capacitors.

It follows from formulas (2) and (3) that, for the same energy stored in the capacitors, the maximum values of the current and power are determined by the exponential—a function of the coefficient \(\beta\). The values of the function for \(\beta\) from \(1\cdot10^{-4}\) to 1 are given in the table. The minimum value of the function

Table

at \(\beta=1\) is equal to \(e^{-2}\approx0.135\). Under all other equal conditions \((W_0,\ R,\ L=\mathrm{const})\), the magnitudes of the maximum current and maximum power increase as the capacitance of the generator decreases.

In the particular case, when the operating voltage is increased by one order of magnitude and the capacitance is decreased by two orders of magnitude, the value of the maximum power increases by a factor of 5.5, while the power rise time decreases correspondingly by one order of magnitude.

It is not difficult to establish that, provided \(C\) and \(L\) are constant, the dependence of the power developed on the load resistance has a maximum. Introducing the notation \(m=\dfrac{R}{\sqrt{L/C}}\) and determining the critical value of \(m\) from the condition \(\dfrac{d}{dm}(P_m)=0\), we have

\[ \frac{\operatorname{tg}^{-1}\sqrt{\dfrac{4}{m^2}-1}} {\sqrt{\dfrac{4}{m^2}-1}} = \frac{m^2+4}{8}, \tag{4} \]

which is valid for \(m=1.118\). The maximum value of the power at the optimum resistance of the discharge circuit is

\[ P_{\mathrm{nom}}=0.65\,\frac{W_0}{\sqrt{LC}}. \]

The optimum value of the resistance is slightly more than half the value of the resistance corresponding to the aperiodicity boundary; therefore there is no need to speak of aperiodization of the process. The decrease in the power developed at \(R>R_{\mathrm{opt}}\) is therefore neither a consequence of breakdown instability at large distances between the electrodes nor a consequence of a decrease in the amount of energy by the moment of discharge, caused by energy losses due to the electrical conductivity of water, as the authors of [2, 7] believe.

The conclusion drawn on the basis of an analysis of the maximum power (3) should be refined according to the condition of the maximum energy released during the first quarter-period:

\[ W^1=\int_0^{t_\phi} i^2(t)R\,dt, \]

where

\[ i=\frac{U}{\omega L}e^{-bt}\sin \omega t,\qquad \phi=\operatorname{arctg}\frac{\omega}{b},\qquad t_\phi=\frac{\phi}{\omega}. \]

With the notation introduced earlier, the amount of energy released during a quarter-period of the current is determined as

\[ W^1 = W_0 \left\{ 1-(1+4\beta) \exp \left[ -2\, \frac{ \operatorname{tg}^{-1}\sqrt{\dfrac{1}{\beta}-1} }{ \sqrt{\dfrac{1}{\beta}-1} } \right] \right\}. \tag{5} \]

From the condition \(\dfrac{dW^1}{d\beta}=0\), the maximum value of the parameter corresponding to the maximum amount of released energy is determined:

\[ \frac{5\beta_{\mathrm{opt}}}{1+4\beta_{\mathrm{opt}}} = \frac{ \operatorname{tg}^{-1}\sqrt{\dfrac{1}{\beta_{\mathrm{opt}}}-1} }{ \sqrt{\dfrac{1}{\beta_{\mathrm{opt}}}-1} }, \tag{6} \]

which corresponds to \(\beta_{\mathrm{opt}}=0.196\).

The dependence of the relative amount of energy released during a quarter-period of the current on the parameter \(\beta\) is presented in Fig. 1. Thus, in cases where the brilliance of the action of an electric discharge in a liquid medium or a solid dielectric is determined mainly by the specific energy density in the spark channel, the capacitance of the generator “in the blow” should be on the order of \(0.2 \varkappa \cdot C_{\mathrm{cr}} = 0.8\,\dfrac{L}{R^2}\). If the determining factor is the rate of energy release, the capacitance of the generator should be less than \(0.8\,\dfrac{L}{R^2}\).

Fig. 1. Dependence of energy release on the parameter \(\beta\).
The lower scale corresponds to the condition of optimality of the parameters.

For an inductance of the discharge circuit of several microhenries and a resistance of the spark channel of \(10\ \Omega\), the value of the optimal capacitance of the generator is several tens of thousands of picofarads.

Until now, the process of energy release after breakdown has been investigated without taking prebreakdown phenomena into account: energy losses before breakdown due to the electrical conductivity of water and the reduction of the amplitude and steepness of the front of the voltage pulse when the generator operates into a load consisting of conducting liquid.

Comparing the effectiveness of using generators at operating voltages of hundreds and tens of kilovolts (for brevity we shall denote them \(U_{100}\) and \(U_{10}\)), we proceed from the equality of the energy losses before breakdown. Assuming that at \(U_{100}\) pulsed breakdown of the gap occurs at a voltage close to the amplitude value, and estimating \(t_{\phi} = 2.3\,\dfrac{L}{R}\) (when an \(L\)—\(R\) circuit is connected to a dc voltage, \(t_{\phi} = 2.3\,\dfrac{L}{R}\) corresponds to \(0.9U_0\), and at the aperiodicity boundary for an \(R\)—\(L\)—\(C\) circuit \(t_{\phi} = 2\,\dfrac{L}{R}\)), the equality of energy losses before breakdown is, in an approximate estimate, described by the equation

\[ \frac{t_{100}}{t_{10}} = 2.3\left(\frac{U_{10}}{U_{100}}\right)^2, \tag{7} \]

where \(t_{100}\) and \(t_{10}\) are the times to breakdown, respectively, at voltages in the hundreds and tens of kilovolts. In the particular case when \(\left(\dfrac{U_{100}}{U_{10}}\right)=10\), \(t_{\phi}=2.3\cdot 10^{-2}t_{10}\). According to data from various authors, the time to breakdown at operating voltages in the tens of kilovolts is several microseconds. If we assume that this time is \(5\) microseconds, then from the equality of the energy losses before breakdown it follows that \(t_{100}=0.1\cdot 10^{-6}\) s, i.e., the time to breakdown at operating voltages in the hundreds of kilovolts must not exceed tenths of a microsecond.

Previously [11] we showed that, with sufficient accuracy for practical application, the output parameters of the voltage wave can be calculated from the generator parameters and the equivalent resistance of the electrode system, measured at high frequency. Estimating the resistance of a multi-electrode electrode system when a technical medium is used as the liquid medium, one may expect a resistance value on the order of \(10^{2}\) ohms. The inductance of the discharge circuit that makes it possible to obtain a voltage pulse with a front duration on the order of \(0.1\) microsecond on a \(100\)-ohm load must not exceed

\[ L=\frac{t_{\phi}\cdot R}{2.3}=4\cdot 10^{-6}\ \text{H}. \]

Impulse-voltage generators for operating voltages in the hundreds of kilovolts in a multistage design have a discharge-circuit inductance of \(10\div 50\) microhenries; that is, the required front duration on a \(100\)-ohm load cannot be obtained with such generators.

Of the various circuits for obtaining high-voltage pulses with a front duration of less than \(10^{-7}\) s, the best is the peaking method when a compensating capacitance is present, comparable in magnitude with the capacitance of the generator. The equivalent circuit of such a generator is shown in Fig. 2, where \(C_{\text{ud}}\) is the capacitance of the generator “in impact,” \(C_{\text{ob}}\) is the compensating capacitance, and \(P_{1}\), \(P_{2}\) are spark gaps, respectively, at the output of the generator and at the output of the compensating capacitance. The voltage on the peaking capacitance, without taking into account the discharge resistance of the generator and the resistance of the spark, is described by the equation

\[ U_{c}=\frac{U_{0}}{1+k}(1-\cos \omega t), \]

Fig. 2. Equivalent calculation circuit of the generator.

where \(U_{0}\) is the total voltage at the generator stages,

\(k\) is a coefficient equal to the ratio \(\dfrac{C_{\text{ob}}}{C_{\text{ud}}}\).

At \(t_{\phi}=\dfrac{\pi}{\omega}\), \(U_{c}=\dfrac{U_{0}}{1+k}\). The energy stored in the compensating capacitance at this moment is \(W_{c}=\dfrac{4k}{(1+k)^{2}}W_{0}\). In the particular case when

\(k = 1,\ U_c = U_0,\ W_c = W_0\), that is, all the energy stored by the generator is at this moment concentrated in the compensating capacitance.

We assume that the spark gap \(P_2\) operates at the moment when the voltage across the compensating capacitance is maximal. The compensating capacitance \(C_{\text{ob}}\), even for \(\kappa < 1\), discharging in the circuit \(L_2 - R\), whose inductance is considerably smaller than the inductance of the circuit \(L_1 - L_2 - R\), provides a substantially greater power than from the discharge of the generator capacitance, whose magnitude we neglect in order to simplify the analysis. The value of the compensating capacitance must be chosen on the basis of the following considerations: increasing the compensating capacitance leads to an increase in the discharge current, but the period of oscillation also increases; as the compensating capacitance increases, with the generator capacitance unchanged, the voltage across the compensating capacitance by the moment of operation of spark gap \(P_2\) decreases. The optimum value of \(\kappa\) may be chosen from the condition of the maximum developed power and the maximum amount of energy released in the spark channel during the first quarter-period of the current.

\[ P_{\max} = \frac{8 W_0 R}{L}\cdot \frac{\kappa}{(1+\kappa)^2} \exp\left[ -2\, \frac{\tg^{-1}\sqrt{\frac{1}{\beta\kappa}-1}} {\sqrt{\frac{1}{\beta\kappa}-1}} \right]. \tag{8} \]

From the condition

\[ \frac{dP_{\max}}{d\kappa}=0 \]

we obtain

\[ \frac{(1-\kappa+2\beta\kappa^2)}{(1+\kappa)} = \frac{\tg^{-1}\sqrt{\frac{1}{\beta\kappa}-1}} {\sqrt{\frac{1}{\beta\kappa}-1}} . \tag{9} \]

The dependence of the maximum power on the coefficient for various values of \(\beta\) is presented in Fig. 3. The solution of the equation of optimality for the coefficient \(\kappa\) is presented in the same figure by the second scale.

Fig. 3. Power nomograms.

Since the values of the discharge circuit and the resistance of the spark channel for a given electrode device are quite definite,

i.e., \(L, R=\mathrm{const}\) for all changes of the parameters \(\beta\) and \(\kappa\), then each value of the generator capacitance “in the impact,” proportional to \(\beta\), corresponds to a definite value of the compensating capacitance, proportional to the coefficient \(\kappa\). As in the absence of a compensating capacitance, the value of the relative developed power over the entire range of \(\kappa\) increases as the generator capacitance decreases. Under the condition that the compensating capacitance is constant, \(\beta\kappa=\mathrm{const}\), the decrease in the relative developed power is determined by the factor \(\dfrac{\kappa}{(1+\kappa)^2}\). To ensure maximum power as the generator capacitance increases, the compensating capacitance must increase much more slowly, since its relative value decreases as the generator capacitance increases.

From an analysis of the expression for maximum power (8) and of the nomogram shown in Fig. 3, it follows that:

1) with the voltage held constant, as the energy stored in the generator capacitors increases \((W_0 \equiv C_y \equiv \beta)\), the power developed in the spark channel increases considerably more slowly, while its relative value decreases;

2) with the compensating capacitance held constant, an excessive increase in the generator capacitance does not lead to an increase in the power developed. Indeed, we do not take into account the power developed in the spark channel during discharge of the generator capacitance in the \(L_1—L_2—R\) circuit, while the power of the compensating capacitance, because the voltage on the compensating capacitance remains unchanged, practically does not increase;

3) for any changes of the parameter \(\beta\), it is not meaningful to take the parameter \(\kappa>1\), since for \(\kappa>1\) both factors in formula 8 decrease. The maximum value of the quantity \(\dfrac{\kappa}{(1+\kappa)^2}\) is determined from the condition

\[ \frac{d}{d\kappa}\left[\frac{\kappa}{(1+\kappa)^2}\right]=0 \quad \text{and at } \quad \kappa=1 \quad \frac{\kappa}{(1+\kappa)^2}=0.25. \]

It has already been indicated above that a justified choice of generator parameters for given values of the inductance of the discharge circuit and the resistance of the spark channel can be made only by jointly considering the conditions of optimality with respect to the maximum developed power and the maximum amount of energy released in the first quarter-period of the current. With the adopted notation,

\[ \frac{W^1}{W_0} = 4\frac{\kappa}{(1+\kappa)^2} \left\{ 1-(1+4\beta\kappa)\exp \left[ -2 \frac{ \operatorname{tg}^{-1}\sqrt{\dfrac{1}{\beta\kappa}-1} }{ \sqrt{\dfrac{1}{\beta\kappa}-1} } \right] \right\}. \tag{10} \]

It is possible to determine the boundary of maximum energy release for optimal parameters \(\beta\) and \(\kappa\). From the condition

\[ \frac{d}{d\beta}\left(\frac{W^1}{W_0}\right)=0 \]

we have

\[ \frac{5\beta\kappa}{1+4\beta\kappa} = \frac{ \operatorname{tg}^{-1}\sqrt{\dfrac{1}{\beta\kappa}-1} }{ \sqrt{\dfrac{1}{\beta\kappa}-1} }. \tag{11} \]

which is valid for \(\kappa = 0.196\). The curve of maximum energy release at the optimum parameters \(\beta\) and \(\kappa\) is presented in Fig. 2 (curve 2) and is marked on the nomogram of energy release (Fig. 4). The parameters corresponding to the optimality condition are presented in these figures by the second scales.

Since within the limits of an oscillatory discharge, i.e., at \(\beta < 1\), the conditions of maximum power and maximum energy release over a quarter-period of current oscillation do not coincide for any value of the parameter \(\beta\), the choice of generator parameters must be determined by the energy characteristics of the processes occurring when the generator discharges into the load.

Fig. 4. Nomograms of energy release.

The analysis presented for the sharpening circuit can be fully transferred to the case in which the sharpening gap \(P_2\) is absent, but a dielectric liquid is used as the liquid medium; that is, the sharpening capacitance is a front capacitance, while the resistance before breakdown is very large. The triggering of the sharpening arrester \(P_2\) for the circuits 2, a, 2, b is equivalent to a change for the circuit 2, c. The greatest effect here can be obtained if the electrode device itself has a considerable capacitance.

In considering questions of power development and energy release in the spark channel during pulsed breakdown in a liquid and in a solid dielectric, we made two assumptions: the resistance of the spark channel was taken, over the entire range of the time considered, as constant and equal to the resistance of the spark channel in the arc stage, and the energy released in the spark channel from the discharge of the generator capacitance in the circuit was not taken into account.

It may be expected that the time at which the resistance of the spark channel decreases to the value

\[ R = 1.118 \sqrt{\frac{L}{C}} \]

will correspond to the premaximum of the power. The criterion for the applicability of the calculation given, under the second assumption, is the condition

\[ \frac{P_{\max}(L_1 - L_2 - R)} {\sqrt{(L_1 + L_2)\, C_{\mathrm{ud}}}} = 0.05\, \frac{P_{\max}(L_2 - R)} {\sqrt{L_2 C_{\mathrm{ob}}}}, \tag{12} \]

that is, the rate of increase of power from the discharge of the generator capacitance in the circuit \(L_1 - L_2 - R\) is negligibly small in comparison with the corresponding value for the circuit \(L_2 - R\).

Conclusions

1. When using pulse-voltage generators to generate shock waves in a liquid medium and in solid dielectrics, increasing the operating voltages from tens to hundreds of kilovolts, even with the same energy stored in the capacitors, makes it possible to achieve higher developed powers and greater energy release in the spark channel.

2. The capacitance of the generator “in the impact” condition, according to the condition of maximum energy release, should not exceed the value \(C_{\mathrm{opt}} = 0.8 \dfrac{L}{R^2}\), which, for a circuit inductance of several microhenries and a spark-channel resistance on the order of 10 ohms, amounts to several tens of thousands of microfarads.

3. When using pulsed discharges in water with operating voltages of hundreds of kilovolts, in order to reduce energy losses before breakdown and to obtain sufficiently high utilization factors of the discharge circuit, it is necessary to employ special circuits for sharpening the voltage pulse with compensating capacitances comparable with the capacitance of the generator “in the impact” condition.

References

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4. L. A. Yutkin. The Electrohydraulic Effect, Mashgiz, 1955.

5. I. M. Astrakhanov. Pressure on a shock wave during a strong spark discharge in water, Neft’—Gaz, No. 10, pp. 87–92, 1959.

6. H. J. Miritz. Hochspannungs—Explosionverformung, ETZ—B, H18, 1964.

7. N. V. Filippov, V. V. Zhurin, V. D. Sulyaev. Inzhenernyi Zhurnal, Nos. 2, 4, 1962.

8. A. I. Limasov, A. T. Chepikov. Volt-second characteristics of polar and nonpolar liquids, Izvestiya SO AN SSSR, 8, 1962.

9. W. D. Edwards. Canad. J. Phys., 1951, 29, 310.

10. A. I. Gavrilin, I. I. Kalyatskii, A. G. Sinebryukhov. Study of the energy characteristics of the breakdown of solid dielectrics, in: Breakdown of Dielectrics and Semiconductors, Energoizdat, 1964.

11. I. I. Kalyatskii, A. F. Usov. Analysis of the operation of a pulse-voltage generator with a conducting-liquid load, in: Breakdown of Dielectrics and Semiconductors, Energoizdat, 1964.