Considering second-harmonic terms in the operation of the phase detector for second-order phase-locked loop

We study the influence of the double-frequency term in the operation of the nonlinear second-order phase-locked loop (PLL). We show that oscillations around the synchronous state obtained from simplified models are unavoidable and we give, in analytical and graphical form, an approximated expression of how the amplitude of the oscillations depends on the normalized loop gain. Numerical simulations confirm the results providing some hints about how to set the PLL gain.