Analysis of the transient dynamics of microwave oscillators

A semi-analytical method for the global prediction and understanding of the transient dynamics of oscillator circuits is presented. It covers both the linear and nonlinear transient stages, which are related with the circuit generalized eigenvalues, here introduced for the first time. The transient model relies on the application of the implicit-function theorem to the harmonicbalance system, in order to derive a reduced-order nonlinear differential equation from a given observation node. This requires the extraction of a nonlinear admittance function, depending on the voltage excitation and oscillation frequency, which is done with a forcing auxiliary generator. The linearization of this admittance function for each excitation amplitude provides a sequence of linear ordinary differential equations, describing the system dynamics in the vicinity of each point of the transient trajectory, which can be reconstructed from the expression of the solution increment at each time step. The sequence of differential equations provides a set of generalized eigenvalues, responsible for the acceleration or deceleration of the oscillation growth and capable to detect spurious transient frequencies. The concept of escape time, or time required by the transient trajectory to go through a certain interval of amplitude values, is also introduced, for the first time to our knowledge. The method has been successfully applied to analyze the transient dynamics of several FET oscillators, including dual-frequency oscillators and switched oscillators.This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF/FEDER) under Project TEC2014-60283-C3-(1/2)-R and Project TEC2017-88242-C3-(1/2)-R

Piecewise quasilinearization techniques for singular boundary-value problems

Abstract Piecewise quasilinearization methods for singular boundary-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions. The accuracy of the globally smooth piecewise quasilinear method is assessed by comparisons with exact solutions of several Lane-Emden equations, a singular problem of non-Newtonian fluid dynamics and the Thomas-Fermi equation. It is shown that the smooth piecewise quasilinearization method provides accurate solutions even near the singularity and is more precise than (iterative) second-order accurate finite difference discretizations. It is also shown that the accuracy of the smooth piecewise quasilinear method depends on the kind of singularity, nonlinearity and inhomogeneities of singular ordinary differential equations. For the Thomas-Fermi equation, it is shown that the piecewise quasilinearization method that provides globally smooth solutions is more accurate than that which only insures global continuity, and more accurate than global quasilinearization techniques which do not employ local linearization

A Newton-Krylov Solution to the Coupled Neutronics-Porous Medium Equations.

The solution of the coupled field equations for nuclear reactor analysis has typically been performed by solving separately the individual field equations and transferring information between fields. This has generally been referred to as “operating splitting” and has been applied to a wide range of reactor steady-state and transient problems. Although this approach has generally been successful, it has been computationally inefficient and imposed some limitations on the range of problems considered. The research here investigated fully implicit methods which do not split the coupled field operators and the solution of the coupled equations using Neutron-Krylov methods. The focus of the work here was on the solution of the coupled neutron and temperature/fluid field equations for the specific application to the high temperature gas reactor. The solution of the neutron field equations was restricted to the steady-state multi-group neutron diffusion equations and the temperature fluid solution for the gas reactor involved the solution of the solid energy, fluid energy, and the single phase mass-momentum equations. In the research performed here, several Newton-Krylov solution approaches have been employed to improve the behavior and performance of the coupled neutronics / porous medium equations as implemented in the PARCS/AGREE code system. The Exact and Inexact Newton's method were employed first, using an analytical Jacobian, followed by a finite difference based Jacobian, and lastly a Jacobian-Free method was employed for the thermal-fluids. Results in the thermal fluids indicate that the Exact Newton's method outperformed the other methods, including the current operator split solution. Finite difference Jacobian and Jacobian-Free were slighty slower than the current solution, though fewer outer iterations were required. In the coupled solution, the exact Newton method performed the best. The finite difference Jacobian with optimized perturbation integrated into the GMRES solve also performed very well, which represented the best iterative solution to the coupled problem. Future analysis will consider the transient problem.Ph.D.Nuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/91609/1/wardam_1.pd