Determination of phase-locked loop statistics with the aid of RLC ladder networks

Transient behavior of the first-order phase-locked loop (PLL) in the presence of noise is investigated. A Fourier series with time varying coefficients is used to represent the modulo-2π phase error probability density function (pdf) in order to solve the Fokker-Planck equation. Solutions for the Fourier coefficients are obtained via simulating
RLC
ladder networks with special element values. Results are shown for cases involving various signal-to-noise (SNR) ratios and initial conditions.