The operation of the phase-lock loop

We examine the operation of the phase-lock loop (PLL). The PLL is one of the fundamental tools of the electrical engineer. It is nonlinear, and its operation is usually examined only after a number of simplifying assumptions are made. We derive an equation that governs this toolʹs operation and determine the properties of the solutions of the equation. We show that the linear time invariant approximation that is generally used to model the PLL is a good approximation to the original nonlinear time-varying equation.