A study on the stability of bipolar-junction-transistor formulation in finite-difference time-domain framework

Abstract

Recently, a new stability result has been put forward for three-dimensional finite-difference time-domain (FDTD) framework by deriving a discrete energy relation similar to the Poynting Theorem in electromagnetism. In this paper, the result is used to show how stability analysis of an FDTD model containing three-terminal nonlinear components can be performed. Here the bipolar junction transistor (BJT) is used to illustrate the idea. It is shown that instability can be due to the BJT contributing numerical energy to the system during simulation, thus causing theand -field components to blow up. This paper also shows how we can prevent instability by identifying the operating region of the device where it is contributing numerical energy. By preventing the device from contributing numerical energy to the system, dynamical stability can be maintained. Formulation, which can source and sink numerical energy, is called conditionally proper. The BJT formulation in FDTD is such an example.

Energy-based stability criterion for the finite-difference time-domain method

“…It should be clearly mentioned that the system matrix A xz is different to the reported ones in [5,6] and carries no special energy distribution. The whole Lyapunov function is summarized into a 15 x 15 square matrix A, where 0 represents a 5 x 5 matrix with zero values.…”

“…As already mentioned, the von Neumann method or the extension via Routh-Hurwitz criterion needs an inevitable transformation process into the fourier or laplace domain, whereas the energy-based stability criterion remains completely in time domain. The versatile approach could be extended to embed the approaches for active [5], lossy or lumped [6] elements seamlessly. The wellknown Courant limits for higher order schemes are computable with proper coefficients, but could not be expressed here due to the lack of space.…”

“…Both methods include an inevitable transformation process where it becomes difficult to apply to subgridding or hybrid schemes. A new approach was proposed by Kung [5], who incorporates discretized Lyapunov functions, but the derived criterion limits the time-step stronger than necessary. An extension which leads to the known Courant limit, were made by Edelvik [6], but his approach could become misleading for higher order FDTD schemes.…”

Composite right/left‐handed extended equivalent circuit (CRLH‐EEC) FDTD: stability and dispersion analysis with examples

“…Recently in Reference [19] a similar approach also based on Liapunov's theorem was used to study the stability of the conventional FDTD including a transistor as a lumped element. These approaches are distinct from the methods that are usually applied to derive stability criteria for FDTD [20].…”

“…For this purpose the time-discrete version of Equation (17) is taken as the starting point. The following step is similar to the time-continuous situation, where the four equations in (17) are multiplied with certain voltages and currents in order to obtain the energies stored in the reactive elements and the losses dissipated by the resistive elements (Equations (19), (20)). For the above-mentioned multiplications the voltages v 0 and v L 1 at t ¼ ðn À 0:5ÞDt and the currents i L 0 and ðÀi 1 Þ at t ¼ nDt; respectively, are used.…”