Extension of the variational equation to analog/digital circuits: numerical and experimental validation

Conventional shooting methods cannot be used to determine the steady-state solution of circuits whose model is characterised by a vector field exhibiting zero order discontinuities. In the analysis of circuits working on a stable limit cycle, this limitation prevents the use of methods that exploit Floquet theory to compute the variational model and thus the properties of the fundamental matrix and, for example, phase noise in oscillators. In this paper, we use an improved shooting method that solves this drawback by resorting to saltation matrices and show how this method makes possible the correct computation of the first left eigenfunction (known as ppv) of the fundamental matrix. ppvs are a key aspect in determining phase noise. The results obtained through numerical simulations are compared with measurements on a relaxation oscillator serving as a simple but significant comparison vehicle.