^{1}, J. Hendriks

^{1}, W. J. M. Brok

^{1}, G. J. H. Brussaard

^{1}and J. J. A. M. van der Mullen

^{1,a)}

### Abstract

In this contribution, a photoconductively switched high-voltage spark gap with an emphasis on the switching behavior is modeled. It is known experimentally that not all of the voltage that is present at the input of the spark gap is switched, but rather a fraction of it drops across the spark gap. This voltage drop depends on the voltage that is present at the input of the spark gap with higher voltages resulting in a smaller drop. We have investigated two possible causes of this: the cathode fall and the resistance of the plasma arc. Using an analytical model of the cathode fall, we have established that the cathode fall can be excluded as the cause of the observed voltage drop. A one-dimensional, time-dependent non-local thermal equilibrium fluid model of the arc plasma has been made. Using this model, the plasma properties have been analyzed for various values of the switched current with emphasis on the conductivity. A good qualitative match between the observed and the simulated dissipation in the gap was found. This indicates that the finite arc resistance is the cause of the observed voltage drop.

The PLASIMO team members, current and former, are acknowledged for their contributions to the code. This work was funded by the Technology Foundation STW, Applied Science Division of NWO and the technology program of the Ministry of Economic Affairs, the Royal Netherlands Academy of Arts and Sciences, and the Foundation for Fundamental Research on Matter (FOM).

I. INTRODUCTION

II. THE SETUP

III. THE CATHODE FALL

A. The cathode fall voltage

B. The cathode fall formation time

C. Conclusion

IV. THE ARC PLASMA

A. The model

B. The initial condition

C. The current

D. Modeling results for

E. Conductivity for various currents

V. CONCLUSION

## Figures

(Color online) The fraction of the voltage that is switched by the spark gap as a function of the input voltage as presented in Ref. 5. For high voltages, almost all the voltage is switched, and only a small fraction of the voltage is lost across the spark gap. With decreasing voltage, an increasing fraction of the voltage is lost across the spark gap. The input voltage is limited to because of self break down.

(Color online) The fraction of the voltage that is switched by the spark gap as a function of the input voltage as presented in Ref. 5. For high voltages, almost all the voltage is switched, and only a small fraction of the voltage is lost across the spark gap. With decreasing voltage, an increasing fraction of the voltage is lost across the spark gap. The input voltage is limited to because of self break down.

(Color online) A schematic drawing of the spark gap switch adapted from Refs. 4 and 5. The gap is an interruption in a coaxial transmission line structure and is filled with either nitrogen or air. By focusing a terawatt laser in a line focus in the spark gap, a laser-produced plasma bridges the gap between the electrodes, switching the voltage that is over the gap.

(Color online) A schematic drawing of the spark gap switch adapted from Refs. 4 and 5. The gap is an interruption in a coaxial transmission line structure and is filled with either nitrogen or air. By focusing a terawatt laser in a line focus in the spark gap, a laser-produced plasma bridges the gap between the electrodes, switching the voltage that is over the gap.

(Color online) Schematic representation of model approximation of the laser-produced plasma. The plasma is assumed to be uniform in the and directions. In the cathode fall model of Sec. III, the plasma is also assumed to be uniform in the direction. In the arc model of Sec. IV, the plasma properties vary along ; the grid in which the numerical calculation of the plasma properties is performed is shown, superimposed on a cross section of the plasma. Symmetry is used, and only the top half of the plasma is simulated. On the top left, the coordinate system, as used throughout this work, is drawn. The size of the slab is based on measurements detailed in Ref. 5. The radius of curvature of the electrodes is , hence, they can be considered as planar on the length scale of the plasma.

(Color online) Schematic representation of model approximation of the laser-produced plasma. The plasma is assumed to be uniform in the and directions. In the cathode fall model of Sec. III, the plasma is also assumed to be uniform in the direction. In the arc model of Sec. IV, the plasma properties vary along ; the grid in which the numerical calculation of the plasma properties is performed is shown, superimposed on a cross section of the plasma. Symmetry is used, and only the top half of the plasma is simulated. On the top left, the coordinate system, as used throughout this work, is drawn. The size of the slab is based on measurements detailed in Ref. 5. The radius of curvature of the electrodes is , hence, they can be considered as planar on the length scale of the plasma.

(Color online) A schematic explanation of the cathode fall formation. The cathode is at , and the anode is at . (a) The femtosecond laser pulse produces a plasma on a time scale that is essentially instantaneous for the plasma we are interested in. (b) The electrons move because of the applied electric field, creating a cathode fall in which the field is compressed. This field facilitates electron emission by the cathode. The heavier ions are essentially motionless on this time scale.

(Color online) A schematic explanation of the cathode fall formation. The cathode is at , and the anode is at . (a) The femtosecond laser pulse produces a plasma on a time scale that is essentially instantaneous for the plasma we are interested in. (b) The electrons move because of the applied electric field, creating a cathode fall in which the field is compressed. This field facilitates electron emission by the cathode. The heavier ions are essentially motionless on this time scale.

(Color online) The field emission current as a function of the electric field at the cathode surface, for values of the work function corresponding to copper and tungsten cathode material. The field emission current of the actual cathode material lies between the two curves in this graph.

(Color online) The field emission current as a function of the electric field at the cathode surface, for values of the work function corresponding to copper and tungsten cathode material. The field emission current of the actual cathode material lies between the two curves in this graph.

(Color online) The electron density in the spark gap as a function of and , for .

(Color online) The electron density in the spark gap as a function of and , for .

(Color online) The electron temperature in the spark gap as a function of and , for . The electron temperature is less relevant outside of the central ionized channel, because is very low there (cf. Fig. 6).

(Color online) The electron temperature in the spark gap as a function of and , for . The electron temperature is less relevant outside of the central ionized channel, because is very low there (cf. Fig. 6).

(Color online) The heavy temperature in the spark gap as a function of and , for .

(Color online) The heavy temperature in the spark gap as a function of and , for .

(Color online) The electrical conductivity in the spark gap as a function of and , for .

(Color online) The electrical conductivity in the spark gap as a function of and , for .

(Color online) The densities of various plasma constituents as a function of .

(Color online) The densities of various plasma constituents as a function of .

(Color online) Left axis: the total resistance of the spark gap plasma at , as a function of . The resistance increases dramatically for decreasing currents. Right axis: the simulated and measured (cf. Ref. 5) percentages of the applied voltage that is switched by the spark gap. The match is good, in particular, for lower currents, although for higher currents, the simulated gap resistance is higher than the observed resistance.

(Color online) Left axis: the total resistance of the spark gap plasma at , as a function of . The resistance increases dramatically for decreasing currents. Right axis: the simulated and measured (cf. Ref. 5) percentages of the applied voltage that is switched by the spark gap. The match is good, in particular, for lower currents, although for higher currents, the simulated gap resistance is higher than the observed resistance.

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