An Analytical Approximation for the Pull-Out Frequency of a PLL Employing a Sinusoidal Phase Detector

ETRI Journal

关键词: separatrix;    focus;    saddle point;    phase portrait;    phase plane;    hyperbolic equilibrium point;    voltage controlled oscillator (VCO);    phase detector (PD);    Phase lock loop (PLL);   

Others  :  1196717
DOI  :  10.4218/etrij.13.0112.0166

【 摘 要 】

The pull-out frequency of a second-order phase lock loop (PLL) is an important parameter that quantifies the loop's ability to stay frequency locked under abrupt changes in the reference input frequency. In most cases, this must be determined numerically or approximated using asymptotic techniques, both of which require special knowledge, skills, and tools. An approximating formula is derived analytically for computing the pull-out frequency for a second-order Type II PLL that employs a sinusoidal characteristic phase detector. The pull-out frequency of such PLLs can be easily approximated to satisfactory accuracy with this formula using a modern scientific calculator.

   

【 参考文献 】

• [1]F.M. Gardner, Phaselock Techniques, 2nd ed., New York: Wiley-Interscience, 1979.
• [2]J.L. Stensby, Phase-Locked Loops: Theory and Applications, Boca Raton, FL: CRC Press, 1997.
• [3]A.-S. Huque and J.L. Stensby, "An Exact Formula for the Pull-Out Frequency of a 2nd-Order Type II Phase Lock Loop," IEEE Commun. Lett., vol. 15, no. 12, Dec. 2011, pp. 1384-1387.
• [4]W.F. Egan, Phase-Lock Basics, Hoboken, NJ: Wiley-Interscience, 1998.
• [5]A.A. Andronov, A.A. Vitt, and S.E. Khaik, Theory of Oscillators, 2nd ed., Mineola, NY: Dover Publications, 1987.
• [6]M.W. Hirsch, S. Smale, and R.L. Devaney, Differential Equations, Dynamical Systems & An Introduction to Chaos, 2nd ed., San Diego, CA: Elsevier Academic Press, 2004.
• [7]L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Berlin, Germany: Springer-Verlag, 1991.
• [8]F.M. Gardner, "Hangup in Phase-Lock Loops," IEEE Trans. Commun., vol. COM-25, no. 10, Oct. 1977, pp. 1210-1214.
• [9]A. Erdelyi, Asymptotic Expansions, Mineola, NY: Dover Publications, 1956.
• [10]F.B. Hildebrand, Introduction to Numerical Analysis, 2nd ed., Mineola, NY: Dover Publications, 1974.
• [11]J. Stensby, "An Approximation of the Pull-Out Frequency Parameter in a Second-Order PLL," Proc. 38th Southeastern Symp. Syst. Theory, 2006, pp. 75-79.