CORRECTION OF THE FRONT OF A HIGH-VOLTAGE NANOSECOND PULSE IN A CIRCUIT WITH A CAPACITIVE LOAD

V. V. KREMNEV

(Presented by the scientific seminar of the Department of Nanosecond Technology)

To obtain high-voltage pulses with a front of the order of \(10^{-9}\) sec (nanoseconds) or less, high-pressure spark gaps are used as switches and sharpeners [1]. The characteristic shape of the pulse front when spark switches are used is shown in Fig. 1a; that is, there is an initial steep section and a gradual transition of the front to the crest. Usually such a pulse is transmitted to the load by means of a system with distributed parameters (cable, strip line). If the load is an active resistance, then no distortion of the pulse front occurs at the load. If the load is capacitive in character (for example, a Kerr cell, inflector plates, etc.), then the pulse front is lengthened; the front duration, determined between the levels \(0.1 \div 0.9\) of the amplitude value, is calculated by the approximate formula

\[ t_{\phi}=\sqrt{t_{\phi 1}^{2}+(2.2RC)^{2}}, \tag{1} \]

where \(t_{\phi 1}\) is the duration of the steep section of the pulse front arriving at the load;

\(R\) is the wave resistance of the transmitting line;

\(C\) is the capacitance of the load.

Fig. 1. Shape of the front of a high-voltage pulse arriving at the load through a transmitting line (a); equivalent circuit (b).

It is of interest to determine the value of \(L\), the inclusion of which causes a shortening of the pulse front, while the amplitude of oscillations at the crest of the pulse does not exceed a certain value.

The equivalent circuit of the resulting network is shown in Fig. 1b. To simplify the calculations, we approximate the curve in Fig. 1a by a dependence of the form

\[ E(t)=E\left[a+\kappa\left(1-e^{-bt}\right)\right]. \tag{2} \]

The quantity \(b\) is then determined as

\[ b=\frac{\ln \dfrac{\varkappa}{0.1}}{t_{0.9}-t_a}, \tag{2a} \]

where \(t_{0.9}\) and \(t_a\) are the times at which the voltage reaches the values \(0.9E\) and \(aE\), respectively (determined from oscillograms).

The voltage across the capacitive load is

\[ \overline{U}_C(\tau)=-\Theta_1 e^{-\frac{\tau}{A}}+\frac{\Theta_3}{\sin\psi}e^{-\delta\tau}\sin(\omega\tau-\psi), \tag{3} \]

where

\[ \Theta_1=\frac{A^2(1-a)}{1-BA+A^2};\qquad \Theta_2=\frac{Aa-A}{1-BA+A^2};\qquad \Theta_3=1+\Theta_2 A; \]

Fig. 2. Graphs of the calculated dependences \(U_{\mathrm{st}}(A)\) for \(B=\mathrm{const}\) (a) and \(B(A)\) for \(\delta=\mathrm{const}\) (b). Dashed curves correspond to minima, solid curves to maxima.

\[ \psi=\operatorname{arcctg}\frac{\Theta_2-\Theta_3\delta}{\Theta_3\omega};\qquad A=\frac{1}{b\sqrt{LC}};\qquad B=\frac{R\sqrt{C}}{\sqrt{L}};\qquad \delta=\frac{B}{2}; \]

\[ \omega=\sqrt{1-\frac{B^2}{4}};\qquad \tau=\frac{t}{\sqrt{LC}};\qquad \overline{U}_C(\tau)=\frac{U_C(\tau)}{2E}. \]

The maximum and minimum values of the voltage at the top are determined from the condition:

\[ \frac{d\overline{U}_C(\tau)}{d\tau}=0. \]

In this case we obtain:

\[ -\sin(\varphi+\psi)e^{\left(-\frac{1}{A}-\delta\right)\tau_M} =\sin(\omega\tau_M-\varphi-\psi), \tag{4} \]

where

\[ \varphi=\operatorname{arctg}\frac{\omega}{\delta}. \]

The most accurate approximation of the curve \(E(t)\) (Fig. 1, a) is obtained for \(a=0.8\) [1]. In Fig. 2, a the graphs of the first two extrema \(U_{Cm}(A)\) for \(B=\mathrm{const}\), calculated by formulas (4) and (3), are given. With the aid of the curves of Fig. 2, a it is not difficult to construct the nomograms of Fig. 2, b, \(B(A)\) for \(\delta=\mathrm{const}\) (\(\delta\) is the magnitude of the extremal deviation in percent of the voltage amplitude on the load). With the aid of the nomograms of Fig. 2, b one can determine the correcting inductance \(L\), if \(b\), \(C\), and \(R\) are known and the permissible magnitude of oscillations at the top of the pulse, in percent of the amplitude, is specified.

Example of calculation

Given: \(C=30\ \text{pF}\), \(b=10^8\ \frac{1}{\text{s}}\), \(R=50\ \Omega\). First determine the quantity

\[ \frac{1}{bRC}:\frac{1}{bRC}=\frac{A}{B}\simeq 6.7. \]

In Fig. 2, b this will be a straight line drawn at the angle \(\varphi_1=\operatorname{arctg}6.7\). On this straight line we choose a point of the plane \(AB\) such that the maximum deviation does not exceed 10% of the pulse amplitude; in this case

\[ B=1.15. \]

Hence

\[ L=\frac{R^2C}{B^2}=57\cdot10^{-9}\ \text{H}. \]

For

\[ \frac{1}{bRC}\gg 7 \]

the correction is not very effective. Then one can increase \(b\) by changing the operating regime of the generator and again carry out an analogous calculation.

In Fig. 3 the dependence \(\bar U_C(\tau)\) is shown for \(\frac{A}{B}=3.3\) at \(L=0\) and \(L=L_{\text{corr}}\). For this case the front, measured between the levels 0.1–0.9, was shortened by a factor of 1.7 due to the correction.

Fig. 3. Curves of voltage rise on a capacitive load for \(\frac{A}{B}=3.3\):
1. \(L=0\), 2. \(L=L_{\text{corr}}\).

An analogous calculation carried out for an ideal unit step at \(a=1\) shows a shortening of the pulse front by approximately a factor of 1.4 under the condition of a 4% overshoot at the top of the pulse. For \(t_{\phi 1}\gg 2.2RC\), it follows from (1) that the correction practically does not reduce the duration of the pulse front.

REFERENCES

1. G. A. Vorob’ev, G. A. Mesyats. Technique for Forming High-Voltage Nanosecond Pulses. Gosatomizdat, 1963.