Magnetostatic Solvers for Conformal Finite Difference Time Domian Particle-In-Cell Simulations: Electromagnets

Magnetic circuit designs are crucial to operations of many vacuum electronics devices and plasma devices, such as magnetrons and magnetically enhanced capacitively coupled plasma (CCP) systems, respectively. For high power microwave (HPM) applications, relativistic magnetrons are one of the most promising HPM sources to deliver desired power efficiently and an electromagnet using coils is commonly employed for providing the required magnetic field for stable operation in a comparably short time duration. Although the conformal finite-difference time-domain (CFDTD) particle-in-cell (PIC) method as implemented in VSim is suitable to model magnetrons and study their electron wave interactions, the magnetic field is usually assumed as uniform fields. On the other hand, for the magnetized CCP, an electromagnetic coil is used to enhance the plasma density and plasma processing efficiency and quality in a CCP discharge. The magnetic field profiles play an important role in the dynamics of charged particles in these systems. However, in most simulations and designs, simple analytic field profiles based on elliptic integrals of an ideal coil are assumed although one may use a third party tool based on a finite element method (FEM) to consider more realistic magnetic field profile. In this work, we propose the magnetostatic (MS) solver newly developed in VSim for modeling an electromagnetic coil as the same finite difference grids are used, avoiding the troubles caused by an export/import process or additional errors due to a translation/interpolation. In the 3-D Cartesian MS solver, the vector potentials in 3-compoent Poisson equations are solved for different configurations of current source distribution such as a single wire, current loop, and Helmholtz coil and the magnetic field profiles obtained are compared with analytic or FEM solutions, giving good agreement. To understand the feasibility of employing the MS solver in the CFDTD PIC simulations, the divergence issues are also evaluated.