Applications of Lyapunov criteria in phase measurements using conformal mapping

DC offsets in sinusoidal voltages affect the phase measurements when the zero crossing method is used. Limitations of the phase shift between ±π/2 set the stability region of the system. Lyapunov criteria are used to examine the system stability based on the Lyapunov theorem. The defined region can be transformed from the complex xy plane to the uv plane where the pulse duration representing the phase is mapped into a positive hyperbolic ellipse where the phase relationship is preserved. The method has been applied to phase-locked loop systems to determine the parameters for stability operation. The method showed that the upper limits of stability and the associated parameters have been qualitatively analyzed in an easy and systematic way