Energy loss to conductors operated at lineal current densities 10 MA=cm: Semianalytic model, magnetohydrodynamic simulations, and experiment

We have developed a semianalytic expression for the total energy loss to a vacuum transmission-line electrode operated at high lineal current densities. (We define the lineal current density j' � B=� 0 to be the current per unit electrode width, where B is the magnetic field at the electrode surface and � 0 is the permeability of free space.) The expression accounts for energy loss due to Ohmic heating, magnetic diffusion, jB work, and the increase in the transmission line's vacuum inductance due to motion of the vacuum-electrode boundary. The sum of these four terms constitutes the Poynting fluence at the original location of the boundary. The expression assumes that (i) the current distribution in the electrode can be approximated as one-dimensional and planar; (ii) the current Iðt Þ¼ 0 for t< 0, and Iðt Þ/ t for t � 0; (iii) j' � 10 MA=cm; and (iv) the current-pulse width is between 50 and 300 ns. Under these con- ditions we find that, to first order, the total energy lost per unit electrode-surface area is given by Wtðt Þ¼ �tB � ðt Þþ �tB � ðtÞ, where BðtÞ is the nominal magnetic field at the surface. The quantities � , � , � , � , � , andare material constants that are determined by normalizing the expression for WtðtÞ to the results of 1D magnetohydrodynamic MACH2 simulations. For stainless-steel electrodes operated at current densities between 0.5 and 10 MA=cm, we find that � ¼ 3:36 � 10 5 , � ¼ 1=2, � ¼ 2, � ¼ 4:47 � 10 4 , � ¼ 5=4, and � ¼ 4 (in SI units). An effective time-dependent resistance, appropriate for circuit simulations of pulsed-power accelerators, is derived from WtðtÞ. Resistance-model predictions are compared to energy-loss measurements made with stainless-steel electrodes operated at peak lineal current densities as high as 12 MA=cm (and peak currents as high as 23 MA). The predictions are consistent with the measurements, to within experimental uncertainties. We also find that a previously used electrode-energy-loss model overpredicts the measurements by as much as an order of magnitude.