Design of Marx generators as a structured eigenvalue assignment

TL;DR

This paper formulates the design of Marx generators as a structured eigenvalue assignment problem, providing symbolic and numerical methods to find solutions for circuits with multiple stages, advancing the design process of pulsed power generators.

Contribution

It introduces a novel structured eigenvalue assignment framework for Marx generator design and compares symbolic and numerical solution approaches for circuits with up to eight stages.

Findings

Symbolic method computes solutions for up to six stages.

Numerical method reaches solutions for up to eight stages.

Conjecture on finitely many solutions for any number of stages remains open.

Abstract

We consider the design problem for a Marx generator electrical network, a pulsed power generator. The engineering specification of the design is that a suitable resonance condition is satisfied by the circuit so that the energy initially stored in a number of storage capacitors is transferred in finite time to a single load capacitor which can then store the total energy and deliver the pulse. We show that the components design can be conveniently cast as a structured real eigenvalue assignment with significantly lower dimension than the state size of the Marx circuit. Then we comment on the nontrivial nature of this structured real eigenvalue assignment problem and present two possible approaches to determine its solutions. A first symbolic approach consists in the use of Gr\"obner basis representations, which allows us to compute all the (finitely many) solutions. A second approach is based on convexification of a nonconvex optimization problem with polynomial constraints. We show that the symbolic method easily provides solutions for networks up to six stages while the numerical method can reach up to seven and eight stages. We also comment on the conjecture that for any number of stages the problem has finitely many solutions, which is a necessary assumption for the proposed methods to converge. We regard the proof of this conjecture as an interesting challenge of general interest in the real algebraic geometry field.