William F. Egan

William F. Egan is an expert and author in the area of PLLs. The first and second editions of his book Frequency Synthesis by Phase Lock[1][2] as well as his book Phase-Lock Basics [3][4] are references among electrical engineers specializing in areas involving PLLs.

Egan's conjecture on the pull-in range of type II APLL
Baseband model of a type II APLL and its closed-form dynamic model

In 1981, describing the high-order PLL, William Egan conjectured that type II APLL has theoretically infinite the hold-in and pull-in ranges.,,,.[1]:176[2]:245[3]:192[4]:161 From a mathematical point of view, that means that the loss of global stability in type II APLL is caused by the birth of self-excited oscillations and not hidden oscillations (i.e., the boundary of global stability and the pull-in range in the space of parameters is trivial). The conjecture can be found in various later publications, see e.g.[5]:96 and[6]:6 for type II CP-PLL. The hold-in and pull-in ranges of type II APLL for a given parameters may be either (theoretically) infinite or empty,[7] thus, since the pull-in range is a subrange of the hold-in range, the question is whether the infinite hold-in range implies infinite pull-in range (the Egan problem[8]). Although it is known that for the second-order type II APLL the conjecture is valid,[9][4]:146 the work by Kuznetsov et al.[8] shows that the Egan conjecture may be not valid in some cases.

A similar statement for the second-order APLL with lead-lag filter is known as Kapranov's conjecture on the pull-in range of type I APLL.[10][11][12] In general, his conjecture is not valid and the global stability and the pull-in range for the type I APLL with lead-lag filters may be limited by the birth of hidden oscillations (hidden boundary of the global stability and the pull-in range).[13][11] For control systems, a similar conjecture was formulated by R. Kalman in 1957 (see Kalman's conjecture).

References
  1. Egan, William F. (1981). Frequency synthesis by phase lock (1st ed.). New York: John Wiley & Sons.
  2. Egan, William F. (2000). Frequency Synthesis by Phase Lock (2nd ed.). New York: John Wiley & Sons.
  3. Egan, William F. (1998). Phase-Lock Basics (1st ed.). New York: John Wiley & Sons.
  4. Egan, William F. (2007). Phase-Lock Basics (2nd ed.). New York: John Wiley & Sons.
  5. Aguirre, S.; Brown, D.H.; Hurd, W.J. (1986). "Phase Lock Acquisition for Sampled Data PLL's Using the Sweep Technique" (PDF). TDA Progress Report. 86 (4): 95–102.
  6. Fahim, Amr M. (2005). Clock Generators for SOC Processors: Circuits and Architecture. Boston-Dordrecht-London: Kluwer Academic Publishers.
  7. Leonov, G. A.; Kuznetsov, N. V.; Yuldashev, M. V.; Yuldashev, R. V. (2015). "Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory". IEEE Transactions on Circuits and Systems I: Regular Papers. IEEE. 62 (10): 2454–2464. arXiv:1505.04262. doi:10.1109/TCSI.2015.2476295.
  8. Kuznetsov, N.V.; Lobachev, M.Y.; Yuldashev, M.V.; Yuldashev, R.V. (2020). "The Egan problem on the pull-in range of type 2 PLLs". IEEE Transactions on Circuits and Systems II: Express Briefs. doi:10.1109/TCSII.2020.3038075.
  9. Viterbi, A. (1966). Principles of coherent communications. New York: McGraw-Hill.
  10. Kapranov M. (1956). "Locking band for phase-locked loop". Radiotekhnika. 2 (12): 37–52.
  11. Kuznetsov, N.V.; Leonov, G.A.; Yuldashev, M.V.; Yuldashev, R.V. (2017). "Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE". Communications in Nonlinear Science and Numerical Simulation. 51: 39–49. Bibcode:2017CNSNS..51...39K. doi:10.1016/j.cnsns.2017.03.010.
  12. Kuznetsov N.V. (2020). "Theory of hidden oscillations and stability of control systems" (PDF). Journal of Computer and Systems Sciences International. 59 (5): 647–668. doi:10.1134/S1064230720050093.
  13. Leonov G.A.; Kuznetsov N.V. (2013). "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits". International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 23 (1): 1330002–219. Bibcode:2013IJBC...2330002L. doi:10.1142/S0218127413300024.
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