Tutorial on dynamic analysis of the Costas loop

This survey is devoted to the dynamic analysis of the Costas loop. In particular the acquisition process is analyzed in great detail. Acquision is most conventiently described by a number of frequency and time parameters such as lock-in range, lock-in time, pull-in range, pull-in time, and hold-in range. While for the classical PLL equations for all these parameters have been derived (many of them are approximations, some even crude approximations), this has not yet been carried out for the Costas loop. It is the aim of this analysis to close this gap. The paper starts with an overview on mathematical and physical models (exact and simplified) of the different variants of the Costas loop, cf. Section 1. In Sections 2–5 equations for the above mentioned key parameters are derived. Finally, the hold-in range of the Costas loop for the case where a lead-lag filter is used for the loop filter is analyzed, cf. Appendix (Section 6).