Analyzing oscillators using multitime PDEs

Abstract

Abstract-Oscillators are often difficult to analyze or simulate, because they generate waveforms that can span a range of widely separated time scales. We present a general oscillator formulation that separates slow and fast dynamics without approximations, and captures amplitude and frequency modulation in a natural and compact manner. To handle frequency-modulation effectively, we make use of a novel concept, warped time, within a multitime partial differential equation framework. The equations incorporate an explicit time-varying frequency variable that matches intuitive notions of changing frequency in a frequency-modulated signal. The formulation is useful for both hand analysis and numerical simulation. Index Terms-Frequency modulation (FM), multitime partial differential equation (MPDE).

Oscillator-AC: restoring rigour to linearized small-signal analysis of oscillators

“…In this section, we first demonstrate the rank-deficiency problem in the DAE/MPDE frequency-domain conversion matrix of oscillators. We then demonstrate how the WaMPDE formulation [10] succeeds in correcting the problem at DC but fails to do so at all other harmonics 1 .…”

“…In the MPDE formulation, signal components with different rates of variation are represented by their "own" artificial time variables [8], [9]. An extension (the Warped MPDE or WaMPDE) of MPDE methods has also been proposed [10] that is able to better analyze amplitude and frequency modulation in oscillators. In the WaMPDE, some time scales are dynamically rescaled (warped time) to make the undulation of FM signal uniform.…”

“…In the WaMPDE, some time scales are dynamically rescaled (warped time) to make the undulation of FM signal uniform. The concept of time warping introduced in the WaMPDE has been further generalized [10] to obtain a family of equation formulations collectively termed the Generalized MPDE, or GeMPDE. A key characteristic of the GeMPDE is that it features "local frequency" variables which are themselves functions of multiple time scales.…”

Fast transient analysis of oscillators using multiple time scales with accurate initial conditions

“…In WaMPDE [4] we must consider at least two time scales: the warped time scale, where the oscillation happens (τ 1 scale) and another time scale equal to the real time (τ 2 scale). The relation between τ 1 and τ 2 is defined by:…”

“…In [3], multiple time axes are employed to separate the fast and slow variations in the response of oscillators, giving a possible way to quickly compute the transient. A variation of that method [4] is known as Warped Multi-time Partial Differential Equation (WaMPDE). The System of ODEs is replaced by a system of partial differential equations.…”