ON THE METHOD FOR SELECTING THE COMPENSATING CAPACITANCE IN A PULSE-SHARPENING SYSTEM

A. F. Usov

Presented by the scientific seminar of the Department of TVN

Recently, in various fields of technology, the use of pulse-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, machining forgings, and other technological operations (1–7).

According to the data of most published works, the operating voltages of the generators amount to tens of kilovolts, and the generator capacitances “in the shot” 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 considerable increase in the power in the discharge and in the rate of its rise can be obtained when 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 exposure times on the order of \(10^{-7}\) sec, the breakdown strength even for extremely nonuniform fields amounts to hundreds of kV/cm (8). In uniform fields, the value \(E_{\mathrm{br}}\) for water at such exposure times, according to (9), may amount even to \(2\) MV/cm. However, when evaluating 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. Considerable energy losses and a low utilization factor of the discharge circuit will also occur when the energy is channeled through a cable, whose characteristic impedance, from the condition of maximum power, should be as small as possible, that is, on the order of 50–100 ohms.

The analysis we carried out earlier showed the necessity of using special voltage-pulse sharpening circuits, the best

of which is the sharpening circuit with a considerable compensating capacitance (10). There we also set forth a method for choosing the magnitude of the compensating capacitance according to the maximum power developed in the breakdown channel and the maximum release of energy during the first quarter-period of the current.

Below a method is presented for choosing the compensating capacitance according to the maximum utilization factor of the discharge circuit.

The overall utilization factor of the discharge circuit, \(k_p\), is determined by the product of the utilization factor of the generator–compensating-capacitance circuit, \(k'_p\), and the utilization factor of the compensating-capacitance–load circuit, \(k''_p\). Without taking into account the damping resistance of the ball gaps of the generator,

\[ k'_p=\frac{2}{1+k}, \quad \text{where } k=\frac{C_{\mathrm{ob}}}{C_{\mathrm{ub}}} \]

is the ratio coefficient of the compensating capacitance to the generator capacitance “in the blow.” With accuracy sufficient for practical use, \(k''_p\) can be calculated as for the circuit \(C_{\mathrm{ob}} — L_{\mathrm{ob}} — R_{\mathrm{n}}\), where

\[ C_{\mathrm{ob}} \text{— compensating capacitance,} \]

\[ L_{\mathrm{ob}} \text{— parasitic inductance of the sharpening circuit,} \]

\[ R_{\mathrm{n}} \text{— load resistance or characteristic impedance of the cable.} \]

For 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 boundary of aperiodicity), we have:

\[ k''_p= \frac{2}{\sqrt{1-\dfrac{1}{\beta k}}} \exp\left[ -\frac{ \ln\left(2\beta k-1+2\beta k\sqrt{1-\dfrac{1}{\beta k}}\right) }{ 2\sqrt{1-\dfrac{1}{\beta k}} } \right] \cdot \operatorname{sh} \frac{ \ln\left(2\beta k-1+2\beta k\sqrt{1-\dfrac{1}{\beta k}}\right) }{2}. \tag{1} \]

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

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

for \(\beta k<1\).

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

The condition for the optimal ratio of the parameters \(\beta\) and \(k\) is determined when

\[ \frac{dk_p}{dk}=0. \]

\[ \frac{\operatorname{arctg}\sqrt{\dfrac{1}{\beta k}-1}} {\sqrt{\dfrac{1}{\beta k}-1}} = \frac{1-k+2\beta k^{2}}{1+k}, \tag{3} \]

for \(\beta k<1\), and

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

Fig. 2. Utilization coefficient of the discharge circuit in the sharpening circuit
in the optimum mode.

\[ \frac{\operatorname{Arth}\sqrt{1-\frac{1}{\beta k}}} {\sqrt{1-\frac{1}{\beta k}}} = \frac{1-k+2\beta k^{2}}{1+k}, \tag{4} \]

for \(\beta k > 1\).

The solution of equations (3) and (4) is represented in Fig. 2 by the second scale. The value of the compensating capacitance is selected from the optimality condition, 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.

In the initial data for constructing the nomograms we did not take into account the damping resistance of the generator sphere gaps, and \(k''_{p}\) was determined without allowance for the parasitic capacitance of the mounting. An experimental verification of the obtained regularities was carried out on

generator of 0.00625 μF with a nominal voltage of 400 kV. The results of the experimental determination of the optimum sharpening regime are given in Fig. 3. Neglecting the damping resistance produces a noticeable discrepancy between the experimental (curve 2) and calculated (curve 1) values of \(k'_p\) only at small values of the compensating capacitance. The error in determining \(k''_p\) does not exceed 5%. Both factors cause a shift of the optimum regime toward larger \(k\). It is reasonable to take a value of the parameter \(k\) somewhat higher than is required by the condition of optimality with respect to the utilization factor of the discharge circuit, since higher values of \(k\) correspond to a larger fraction of the energy stored in the compensating capacitance by the instant at which the sharpening spark gap fires. The efficiency of recharging is

\[ \eta_1=\frac{4k}{(1+k)^2}, \]

i.e., already at \(k \approx 0.5\) practically all the energy stored by the generator by the instant at which the sharpening spark gap fires is concentrated in the compensating capacitance. The dependence of the recharging efficiency on the capacitance ratio is presented in Table 1.

Table 1

It was already pointed out above that ordinary multistage generators 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 2–2.5 μH. Therefore, even when \(k\) is considerably 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 specific nature of the processes occurring during pulsed discharge in a liquid medium or a solid dielectric, it is sometimes necessary to place the voltage source at considerable distances. It was of interest to consider the operating conditions of the generator when the energy is transmitted through a cable. Regardless of the load resistance at the end of the cable, \(k_p\) for the beginning of the cable is determined by the discharge of the generator into the surge impedance of the cable, which, as was indicated above, may have a value of the order of 50–100 ohms. 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 a low efficiency. If no stringent requirements are imposed on the waveform, then \(k_p\) at the beginning of the cable can be considerably increased by connecting a front capacitance at the cable input. The ratio of \(k_p\) at the beginning and at the end of the cable (Fig. 4) strictly obeys the relation:

\[ U_{\mathrm{н}}=\alpha U_{\mathrm{к}}. \]

where:

\[ U_{\mathrm{н}} \text{ — voltage across the load,} \]

\[ U_{\mathrm{к}} \text{ — voltage at the cable input,} \]

\[ \alpha \text{ — refraction coefficient } \alpha=\frac{2R_{\mathrm{н}}}{z_{\mathrm{в}}+R_{\mathrm{н}}}, \]

Fig. 3. Experimental determination of the optimal value of \(\kappa\).

1 — \(\kappa'_p\) — without allowance for \(R_i\) (calculation),
2 — \(\kappa'_p\) with allowance for \(R_i\) (experimental),
3 — \(\kappa''_p\) — for the sharpening circuit \((R_{\mathrm{n}} = 175\ \Omega)\),
4 — \(\kappa_p := \kappa'_p \times \kappa''_p\) — overall utilization coefficient of the discharge circuit.

$R_{\mathrm{H}}$ — load resistance,

$z_{\mathrm{B}}$ — characteristic impedance of the cable ($z_{\mathrm{B}} = 75$ ohms). For the case of dielectric liquids, $\eta \approx 2$.

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 the transition from dielectric liquids to technical water causes a decrease in $\alpha$ 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 there arises a need for its continuous purification. If the parameters of the sharpening circuit make it possible to obtain a sufficiently high utilization coefficient, then the permissible degree of chemical

Fig. 4. Utilization coefficient of the discharge circuit for the beginning (1) and end of the cable (2) at different load resistances, $z_{\mathrm{B}} = 75$ ohms.

contamination of the liquid medium can be increased by reducing the characteristic impedance of the cable. The dependences obtained in the analysis of the sharpening circuit can also be used in selecting the optimal ratios between the load resistance and the characteristic impedance of the cable. Indeed, if $\beta$ is understood as the ratio

\[ \frac{C_{\mathrm{ob}}}{C_{\mathrm{kr}}} \left( C_{\mathrm{kr}} = \frac{4L_{\mathrm{ob}}}{R_{\mathrm{H}}^{2}} \right), \]

and $k$ is understood as

\[ \frac{z_{\mathrm{B}}}{R_{\mathrm{H}}}, \]

then

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

and the initial formulas for determining the optimal ratios of $R_{\mathrm{H}}$ and $z_{\mathrm{B}}$ are analogous to those considered earlier for the sharpening circuit. By jointly solving the equations for the individual sections of the circuit, it is possible, by the nomographic method, to express the optimal ratios among all elements of the circuit.

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

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