Stray Capacitance in a Plasma Focus Device: Implications on the Current Derivative Calibration and the Effective Discharge Current

Abstract

The notion that PF discharge circuits should be represented by an equivalent circuit having two loops instead of the traditional single one is presented. This implies that two frequencies must be expected in the currents and voltages in these devices. Also, that the current flowing into the plasma is not the same as the current flowing from the capacitor bank. Finally, the difficulties for calibrating in situ a Rogowski coil are discussed.

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Fig. 1
Fig. 2
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Correspondence to M. Barbaglia.

Appendix

Appendix

The equations ruling the behavior of the circuit given in Fig. 2 in the short-circuit configuration are:

$$\left\{ {\begin{array}{*{20}l} {L_{2} \dot{I}_{2} - \frac{{Q_{p} }}{{C_{p} }} = 0} \hfill \\ {L_{1} \dot{I} - \frac{{Q_{0} }}{{C_{0} }} + \frac{{Q_{p} }}{{C_{p} }} = 0} \hfill \\ {\dot{Q}_{0} = I} \hfill \\ {\dot{Q}_{p} = I_{1} = I - I_{2} } \hfill \\ \end{array} } \right.$$

This can be rearranged as:

$$\begin{array}{*{20}l} {\left\{ {\begin{array}{*{20}l} {\ddot{Q}_{0} + \omega_{1}^{2} Q_{0} + \omega_{3}^{2} Q_{p} = 0} \hfill \\ {\ddot{Q}_{p} + \omega_{1}^{2} Q_{0} + \omega_{2}^{2} Q_{p} = 0} \hfill \\ \end{array} } \right.} \hfill \\ {\omega_{1}^{2} = \frac{1}{{L_{1} C_{0} }}} \hfill \\ {\omega_{3}^{2} = \frac{1}{{L_{1} C_{p} }}} \hfill \\ {\omega_{2}^{2} = \left( {\frac{1}{{L_{1} }} + \frac{1}{{L_{2} }}} \right)\frac{1}{{C_{p} }}} \hfill \\ \end{array}$$

Their solutions can be written as linear combinations of sin(λt) and cos(λt) where the eigenvalues λ are obtained from:

$$ \left| \begin{array}{lr} \omega_{1}^{2} - \lambda^{2} & \omega_{3}^{2} \\ \omega_{1}^{2} & \omega_{2}^{2} - \lambda^{2} \end{array} \right| =0$$

That is

$$\lambda^{4} - \left( {\omega_{1}^{2} + \omega_{2}^{2} } \right)\lambda^{2} + \omega_{1}^{2} \left( {\omega_{2}^{2} - \omega_{3}^{2} } \right) = 0$$

this yields

$$\lambda_{ \pm }^{2} = \frac{1}{2}\left[ {\left( {\omega_{1}^{2} + \omega_{2}^{2} } \right) \pm \sqrt {\left( {\omega_{1}^{2} + \omega_{2}^{2} } \right)^{2} - 4\omega_{1}^{2} \left( {\omega_{2}^{2} - \omega_{3}^{2} } \right)} } \right]$$

In the limit C p  << C o , which is satisfied in all PF devices, the problem becomes simpler and after a little algebra it can be shown to become

$$\lambda_{ + }^{2} \; \approx \;\frac{{L_{1} + L_{2} }}{{C_{p} L_{1} L_{2} }} ;\quad \lambda_{ - }^{2} \; \approx \;\frac{1}{{C_{0} \left( {L_{1} + L_{2} } \right)}}$$

The resulting frequencies are easily interpreted in terms of simple circuit theory. The root \(\lambda_{ - }\) (=2π/T +, the larger time period) corresponds to the frequency expected from an LC circuit with a single capacitor (C o ) and the inductances L 1 and L 2 in series (that is, what should be expected if the impedance of C p were infinite, a reasonable approximation for the lower frequency case) while \(\lambda_{ + }\) (=2π/T , the smaller time period) is the root corresponding to a circuit with a single capacitor (C p ) and both inductances in parallel, that is, just as if C o have been replaced by a short circuit, also a reasonable approximation for the higher frequency case.

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Bruzzone, H., Acuña, H., Barbaglia, M. et al. Stray Capacitance in a Plasma Focus Device: Implications on the Current Derivative Calibration and the Effective Discharge Current. J Fusion Energ 36, 87–91 (2017). https://doi.org/10.1007/s10894-017-0126-1

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Keywords

  • Plasma focus
  • Pinch
  • Plasma measurements
  • Self-compression