The Dilemma of Resolving the Low-Frequency Breakdown Problem in Microwave Components via Traditional and Improved Finite-Element Time-Domain Techniques

This paper developed an improved numerical methods for calculating electromagnetic field at low frequencies, and compared the performance with the traditional method. Traditional finite-element time-domain (FETD) is known to suffer from low frequency breakdown (LFB) problem, where system matrix becomes ill-conditioned, and leads to instability of small size electrical structures. In this paper, we proposed a mixed FETD (mFETD) method in resolving the LFB anomaly in the traditional FEM, with a particular reference to transmission line, inductor, and coaxial cable. We considered wave equation of E-field with incorporation of divergence constraint equation (DCE) as a function of Lagrange multiplier using current continuity equation (CCE) and Gauss’s law. For spatial discretization, both nodal basis functions and Curl conforming vector basis functions were selected, and we employed implicit Newmark beta algorithm (NBA) for integration of time. We describe how the components constructed via DCE in system matrix mitigate the singularity impact of the stiffness matrix, which consequently led to significant improvement of the system matrix. Numerical experiment and results demonstrate how the mFETD obtains stable numerical solution and attains faster rate of convergence while using an iterative solver, as such, improves the computational efficiency. Therefore, the mFETD method handles the transient problems in transmission line, which causes LFB anomaly in the traditional FETD, but not efficient in terms of computational time and iteration at solving the coaxial cable problem.