Finite element computation of a viscous compressible free shear flow governed by the time dependent Navier-Stokes equations

A finite element algorithm for solution of fluid flow problems characterized by the two-dimensional compressible Navier-Stokes equations was developed. The program is intended for viscous compressible high speed flow; hence, primitive variables are utilized. The physical solution was approximated by trial functions which at a fixed time are piecewise cubic on triangular elements. The Galerkin technique was employed to determine the finite-element model equations. A leapfrog time integration is used for marching asymptotically from initial to steady state, with iterated integrals evaluated by numerical quadratures. The nonsymmetric linear systems of equations governing time transition from step-to-step are solved using a rather economical block iterative triangular decomposition scheme. The concept was applied to the numerical computation of a free shear flow. Numerical results of the finite-element method are in excellent agreement with those obtained from a finite difference solution of the same problem

Rigorous Stability Criterion for Digital Phase Locked Loops

This paper proposes a rigorous stability criterion for an arbitrary order digital phase locked loop (DPLL), with a charge pump phase frequency detector (CP-PFD) component. Stability boundaries for such systems are determined using piecewise linear methods to model the nonlinear nature of the CP-PFD component block. The model calculates the control voltage, after a predetermined number of input reference signal sampling periods, to a small initial voltage offset. This paper, in particular, takes an in-depth look at the second order system. The second order stability boundaries, as defined by the proposed technique, are compared to that of existing linear theory stability boundaries, and display a significant improvement. The applicability of the proposed technique to higher order systems, using a numerically iterative solution, is presented. Finally the proposed methodology is used to determine the stability boundary of a third order system and thus the component values for a stable system. Using these component values the response of the DPLL to an initial control voltage offset is simulated using a circuit level simulation. Index Terms—High Order, Phase Locked Loop, Piecewise Linear, Stability

Rigorous Stability Criterion for Digital Phase Locked Loops

This paper proposes a rigorous stability criterion for an arbitrary order digital phase locked loop (DPLL), with a charge pump phase frequency detector (CP-PFD) component. Stability boundaries for such systems are determined using piecewise linear methods to model the nonlinear nature of the CP-PFD component block. The model calculates the control voltage, after a predetermined number of input reference signal sampling periods, to a small initial voltage offset. This paper, in particular, takes an in-depth look at the second order system. The second order stability boundaries, as defined by the proposed technique, are compared to that of existing linear theory stability boundaries, and display a significant improvement. The applicability of the proposed technique to higher order systems, using a numerically iterative solution, is presented. Finally the proposed methodology is used to determine the stability boundary of a third order system and thus the component values for a stable system. Using these component values the response of the DPLL to an initial control voltage offset is simulated using a circuit level simulation. Index Terms—High Order, Phase Locked Loop, Piecewise Linear, Stability