On a Method for Selecting Elements of a Circuit for Sharpening the Front of High-Voltage Pulses

I. I. Kalyatskii, A. F. Usov

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

Recently, in various fields of technology, pulse-voltage generators have been widely used to generate shock waves in liquid media and solid dielectrics for the purpose of destroying solid bodies, for the kinetics of chemical reactions that usually proceed at high pressures and temperatures, for stamping, processing forgings, and other technological operations [1–7].

According to most published works, the operating voltages of the generators amount to tens of kilovolts, and the capacitances of the generators “in impact” are 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 breakdown channel.

A significant increase in the power in the discharge and in the rate of its rise can be obtained by using pulse-voltage generators of hundreds of kilovolts with generator capacitances of several tens of thousands of picofarads. The possibility of using operating voltages of hundreds of kilovolts is due to the fact that, for voltage-pulse action times on the order of \(10^{-7}\) sec, the breakdown strength even for extremely nonuniform fields is hundreds of kV/cm [8]. In uniform fields, the value \(E_{\mathrm{br}}\) for water at such action times, according to [9], can be as high as 2 kV/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, in a multistage design, such generators possess a parasitic inductance of the discharge circuit of tens of microhenries; that is, when water or an electrolyte is used as the liquid medium, considerable energy losses will occur.

Often, owing to operating conditions, the generator may be located at a distance of tens and even hundreds of meters from the load. In this case as well there will be considerable energy losses and a low utilization factor of the discharge circuit when the generator operates into the surge impedance of the cable, whose value, from the condition of maximum power, should be as small as possible, i.e., on the order of 50–100 ohms.

The analysis we carried out earlier [10] showed the need to use special voltage-pulse sharpening circuits, the best of which is a sharpening circuit with a substantial compensating capacitance [11–12]. There we also set forth a method for selecting the value of the

of the compensating capacitance in terms of the maximum power developed in the channel gap and the maximum energy release during the first quarter-period of the current.

Below a procedure is set forth for selecting the compensating capacitance and the wave resistance of the cable according to the maximum utilization coefficient of the discharge circuit.

The total utilization coefficient of the discharge circuit \(\kappa_p\) is determined by the product of the utilization coefficient of the generator--compensating-capacitance circuit \(\kappa'_p\) and the utilization coefficient of the compensating-capacitance--load circuit \(\kappa''_p\). Without taking into account the damping resistance of the generator spark gaps,

\[ \kappa'_p=\frac{2}{1+\kappa}, \]

where \(\kappa=\dfrac{C_{\mathrm{ob}}}{C_{\mathrm{ud}}}\) is the ratio coefficient of the compensating capacitance and the generator capacitance “in the shock.” With accuracy sufficient for practical use, \(\kappa''_p\) can be calculated, since for the circuit \(C_{\mathrm{ob}}-L_{\mathrm{ob}}-R_{\mathrm{n}}\), where

\(C_{\mathrm{ob}}\) is the compensating capacitance,

\(L_{\mathrm{ob}}\) is the parasitic inductance of the sharpening circuit,

\(R_{\mathrm{n}}\) is the load resistance or the wave resistance of the cable.

With the introduced parameter \(\beta=\dfrac{C_{\mathrm{ud}}}{C_{\mathrm{kr}}}\) \(\left(C_{\mathrm{kr}}=\dfrac{4L}{R^2}\right.\) is the value of the compensating capacitance corresponding to the aperiodicity boundary\()\), we have

\[ \kappa''_p= \frac{2}{\sqrt{1-\dfrac{1}{\beta\kappa}}} \exp\left[ -\frac{\operatorname{Arth}\sqrt{1-\dfrac{1}{\beta\kappa}}} {\sqrt{1-\dfrac{1}{\beta\kappa}}} \right] \operatorname{sh}\left( \operatorname{Arth}\sqrt{1-\dfrac{1}{\beta\kappa}} \right) \tag{1} \]

for \(\beta\kappa>1\), and

\[ \kappa''_p= \frac{2}{\sqrt{\dfrac{1}{\beta\kappa}-1}} \exp\left[ -\frac{\operatorname{arctg}\sqrt{\dfrac{1}{\beta\kappa}-1}} {\sqrt{\dfrac{1}{\beta\kappa}-1}} \right] \sin\left( \operatorname{arctg}\sqrt{\dfrac{1}{\beta\kappa}-1} \right) \tag{2} \]

where \(\beta\kappa<1\).

A nomogram of the utilization coefficient of the generator discharge circuit with sharpening is given in Fig. 1.

The condition for the optimal ratio of the parameters \(\beta\) and \(\kappa\) is determined from

\[ \frac{d\kappa_p}{d\kappa}=0. \]

\[ \frac{ \operatorname{Arth}\sqrt{1-\dfrac{1}{\beta\kappa}} }{ \sqrt{1-\dfrac{1}{\beta\kappa}} } = \frac{1-\kappa+2\beta\kappa^2}{1+\kappa} \tag{3} \]

for \(\beta\kappa>1\), and

\[ \frac{ \operatorname{arctg}\sqrt{\dfrac{1}{\beta\kappa}-1} }{ \sqrt{\dfrac{1}{\beta\kappa}-1} } = \frac{1-\kappa+2\beta\kappa^2}{1+\kappa} \tag{4} \]

for \(\beta\kappa<1\).

The solution of equations (3) and (4) is presented in Fig. 1 by the second scale. The value of the compensating capacitance is selected from the condition of optimality for the specified values of the generator and load capacitances and for the minimum inductance of the sharpening circuit attainable with the given method of mounting the compensating capacitance, the sharpening spark gap, and the load or cable.

Fig. 1. Nomogram of the utilization coefficient of the discharge circuit in the sharpening circuit.

In the initial data for constructing the nomograms we did not take into account the damping resistance of the generator ball gaps, and we determined \(\kappa_p^{\prime\prime}\) without taking into account the parasitic capacitance of the mounting. An experimental check of the regularities obtained was carried out on a generator of \(0.00625\ \mu\mathrm{F}\) with a rated voltage of \(400\ \mathrm{kV}\). The results of the experimental determination of the optimal sharpening regime are given in Fig. 2. Neglect of the damping resistance causes a noticeable discrepancy between the experimental (curve 2) and calculated (curve 1) values of \(\kappa_p^{\prime\prime}\) only at small values of the compensating capacitance. The error in determining \(\kappa_p^{\prime\prime}\) does not exceed 5%. Both of these factors cause a shift of the optimal regime toward larger \(\kappa\). If the parameters of the circuit make it possible to obtain a utilization coefficient higher than is necessary for the given interelectrode gap, it makes sense to take a somewhat overestimated value of the parameter \(\kappa\) than is required by the condition of optimality with respect to the utilization coefficient of the discharge circuit, since larger values of \(\kappa\) correspond to a larger fraction of stored

in the compensating capacitance by the moment of operation of the sharpening spark gap. The recharge efficiency is \(\eta=\dfrac{4\kappa}{(1+\kappa)^2}\), i.e., already at \(\kappa=0.5\) practically all the energy stored by the generator by the moment of operation of the sharpening spark gap is concentrated in the compensating capacitance. The dependence of the recharge efficiency on the capacitance ratio is presented in Table 1.

Table 1

It has already been pointed out above that ordinary multistage generators rated for hundreds of kilovolts have a discharge-circuit inductance of tens of microhenries. For the sharpening circuit it is quite feasible to limit the circuit inductance to a value of \(2\)—\(2.5\ \mu\text{H}\). Therefore, even for \(\kappa\) less than 1, the power developed in the sharpening circuit exceeds the corresponding value for the generator—load circuit in the absence of sharpening.

Because of the specificity of the processes occurring during pulsed discharge in a liquid medium or a solid dielectric, it is sometimes necessary to move the voltage source to considerable distances. It was of interest to consider the question of the operating conditions of the generator when the energy is channeled through a cable. Independently of the load resistance at the end of the cable, \(\kappa_r\) for the beginning of the cable is determined by the discharge of the generator into the wave impedance of the cable, which, as already indicated above, may be of the order of \(50\)—\(100\ \Omega\). It is clear that the use of generators with operating voltages of hundreds of kilovolts without employing a sharpening circuit is associated with large energy losses and low efficiency. If no stringent requirements are imposed on the waveform, then \(\kappa\) at the beginning of the cable can be substantially increased by connecting a front capacitance to the cable input. The ratio of \(\kappa_r\) at the beginning and at the end of the cable is strictly governed by the relation

\[ U_2 = a U_1, \]

Fig. 2. Experimental determination of the optimum value: 1 — \(p\) without allowance for \(R_i\) (calculation), 2 — \(k_p\) with allowance for \(R_i\) (experimentally), 3 — \(k'_p\) for the sharpening circuit \((R_n = 175\ \Omega)\), 4 — \(k_p = k'_p \times k''_p\) — the overall utilization coefficient of the discharge circuit.

where \(U_2\) is the voltage across the load,

\(U_1\) is the voltage at the cable input,

\(\alpha\) is the reflection coefficient,

\[ \alpha=\frac{2R_{\mathrm{n}}}{R_{\mathrm{n}}+z_{\mathrm{B}}}, \]

\(R_{\mathrm{n}}\) is the load resistance,

\(z_{\mathrm{B}}\) is the characteristic impedance of the cable.

For dielectric liquids, \(\alpha=2\).

If the load resistance, even when technical water and electrolytes are used, exceeds the characteristic impedance of the cable by a factor of 2–3, then replacing dielectric liquids by technical water causes \(\alpha\) to decrease by less than a factor of 1.5. This is especially important if the technological process is associated with large leaks of the liquid medium, or if the liquid medium is subjected to chemical action and it becomes necessary to purify it continuously. If the parameters of the sharpening circuit make it possible to obtain a sufficiently high utilization factor, then the permissible degree of chemical contamination of the liquid medium can be increased by reducing the characteristic impedance of the cable.

The dependences obtained in the analysis of the circuit with sharpening can also be used in choosing the optimal ratios between the load resistance and the characteristic impedance of the cable. Indeed, if \(\beta\) is understood as the ratio

\[ \frac{C_{\mathrm{ud}}}{C_{\mathrm{kr}}}\left|\, C_{\mathrm{kr}}=\frac{4L}{R_{\mathrm{n}}^{2}}\,\right|, \quad \text{and } \kappa \]

is understood as

\[ \gamma=\frac{z_{\mathrm{B}}}{R_{\mathrm{n}}}, \quad \text{then} \]

\[ \alpha=\frac{2R_{\mathrm{n}}}{z_{\mathrm{B}}+R_{\mathrm{n}}} =\frac{2}{1+\gamma}, \]

and the initial formulas for determining the optimal ratios of \(R_{\mathrm{n}}\) and \(z_{\mathrm{B}}\) are similar to those considered earlier for the circuit with sharpening.

The utilization factor of the discharge circuit generator—cable—load is:

\[ \kappa_{\mathrm{p}}= \frac{2}{1+\gamma}\cdot \frac{2}{\sqrt{1+\frac{1}{\beta\gamma^{2}}}}\, \exp\left[ -\frac{\operatorname{Arth}\sqrt{1-\frac{1}{\beta\gamma^{2}}}} {\sqrt{1-\frac{1}{\beta\gamma^{2}}}} \right] \times \]

\[ \times\, \operatorname{sh}\left( \operatorname{Arth}\sqrt{1-\frac{1}{\beta\gamma^{2}}} \right) \tag{5} \]

for \(\beta\gamma^{2}>1\), and

\[ \kappa_{\mathrm{p}}= \frac{2}{1+\gamma}\cdot \frac{2}{\sqrt{\frac{1}{\beta\gamma^{2}}-1}}\, \exp\left[ -\frac{\operatorname{arctg}\sqrt{\frac{1}{\beta\gamma^{2}}-1}} {\sqrt{\frac{1}{\beta\gamma^{2}}-1}} \right] \times \]

\[ \times\, \sin\left( \operatorname{arctg}\sqrt{\frac{1}{\beta\gamma^{2}}-1} \right) \tag{6} \]

for \(\beta\gamma^{2}<1\).

The condition for the optimal ratio of the parameters \(\beta\) and \(\gamma\) is determined by

\[ \frac{d k_p}{d\gamma}=0. \]

\[ \frac{ \operatorname{Arth}\sqrt{1-\dfrac{1}{\beta\gamma^2}} }{ \sqrt{1-\dfrac{1}{\beta\gamma^2}} } = \frac{1+\beta\gamma^3}{1+\gamma} \tag{7} \]

for \(\beta\gamma^2>1\), and

\[ \frac{ \operatorname{arctg}\sqrt{\dfrac{1}{\beta\gamma^2}-1} }{ \sqrt{\dfrac{1}{\beta\gamma^2}-1} } = \frac{1+\beta\gamma^3}{1+\gamma} \quad \text{for } \beta\gamma^2<1. \tag{8} \]

The nomogram of the utilization coefficient and the solution of the equations of the optimal ratio of the parameters \(\beta\) and \(\alpha\) are presented in Fig. 3.

The solution of the optimality condition in a sharpening circuit in the presence of the cable wave impedance assumes two cases:

1) selection of the value of the compensating capacitance for given \(\beta\) and \(\gamma\), i.e., the generator and the load are specified, and the wave impedance is chosen arbitrarily;

2) selection of the value of the cable wave impedance for given \(\beta\) and \(\kappa\), i.e., the generator and the load are specified, and the value of the compensating capacitance is chosen arbitrarily.

For the first case the solution is obtained from the condition

\[ \frac{d k_p}{d\kappa}=0, \]

i.e., for

\[ \beta\gamma^2\kappa>1 \]

\[ \frac{ \operatorname{Arth}\sqrt{1-\dfrac{1}{\beta\gamma^2\kappa}} }{ \sqrt{1-\dfrac{1}{\beta\gamma^2\kappa}} } = \frac{1-\kappa+2\beta\gamma^2\kappa^2}{1+\kappa}. \tag{9} \]

Fig. 3. Nomogram of the utilization coefficient of the discharge circuit generator—cable—load.

and for \(\beta\gamma\kappa < 1\)

\[ \frac{ \operatorname{arctg}\sqrt{\dfrac{1}{\beta\gamma^{2}\kappa}-1} }{ \sqrt{\dfrac{1}{\beta\gamma^{2}\kappa}-1} } = \frac{1-\kappa+2\beta\gamma^{2}\kappa^{2}}{1+\kappa}. \tag{10} \]

For the second case the solution is obtained from the condition \(\dfrac{d\kappa_p}{d\gamma}=0\), i.e., for \(\beta\gamma^{2}\kappa > 1\)

\[ \frac{ \operatorname{Arth}\sqrt{1-\dfrac{1}{\beta\gamma^{2}\kappa}} }{ \sqrt{1-\dfrac{1}{\beta\gamma^{2}\kappa}} } = \frac{1+\beta\gamma^{3}\kappa}{1+\gamma} \tag{11} \]

Fig. 4. Nomogram of the utilization coefficient of the discharge circuit in a sharpening circuit for arbitrary choice of the cable wave resistance value.

and for \(\beta\gamma^{2}\kappa < 1\)

\[ \frac{ \operatorname{arctg}\sqrt{\dfrac{1}{\beta\gamma^{2}\kappa}-1} }{ \sqrt{\dfrac{1}{\beta\gamma^{2}\kappa}-1} } = \frac{1+\beta\gamma^{3}\kappa}{1+\gamma}. \tag{12} \]

Fig. 5. Nomogram of the utilization coefficient of the discharge circuit in the sharpening circuit with a cable, for an arbitrary choice of the value of the compensating capacitance.

Fig. 6. On the method for selecting optimal parameter ratios in the sharpening circuit with a cable.

The corresponding nomograms are presented in Figs. 4 and 5. The simultaneous solution of equations 9, 10, 11, and 12, and the scheme for selecting the optimal parameters \(\kappa\) and \(\gamma\), are presented in Fig. 6. For a specified value of \(\beta\), \(\kappa_{\mathrm{opt}}\), \(\gamma_{\mathrm{opt}}\), and the maximum attainable utilization coefficient of the discharge circuit for the given \(\beta\) are determined.

The use of the optimal ratios obtained in selecting the parameters of the individual elements of the circuits of electropulse installations is a most important reserve for improving their techno-economic performance.

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