Minimal inductance configurations for radially dependent transmission lines

To maximize the current delivered by pulsed power generators, such as the Sandia Z machine, it is advantageous to employ transmission lines possessing the smallest inductance compatible with current insulation requirements. We extend previous workl for constant gap axisymmetric transmission lines to include the case of radially dependent gaps. Following Ref. 1 we solve the Euler-Lagrange equation to obtain the mid-gap z(r) curve yielding minimal inductance for the prescribed radial dependence of the gap w(r), and given end-points as boundary conditions. For the linear case w(r)=wO+kr the mid-gap curve is given analytically, up to one parameter: the mid-gap radius of curvature at the inner endpoint, which is determined by solving numerically the resulting algebraic equation. We also present the numerical integration of the single ordinary equation obtained from the Euler-Lagrange equation for w(r) being a function other than linear. We provide examples of minimum inductance lines and their applications. We also examine the question of the connection of a given minimal inductance line with the load and feed line, given that the optimization procedure presented in this work does not allow for arbitrary line terminations: The anode and cathode surfaces, including their end-points are completely determined by the boundary conditions and the gap radial dependence.