Renat V. Yuldashev

Analytical Method for Computation of Phase-Detector Characteristic

Discovery of undesirable hidden oscillations, which cannot be found by simulation, in models of phase-locked loop (PLL) showed the importance of development and application of analytical methods for the analysis of such models. Approaches to a rigorous nonlinear analysis of analog PLL with multiplier phase detector (classical PLL) and linear filter are discussed. An effective analytical method for computation of multiplier/mixer phase-detector characteristics is proposed. For various waveforms of high-frequency signals, new classes of phase-detector characteristics are obtained, and dynamical model of PLL is constructed.

Hold-In, Pull-In, and Lock-In Ranges of PLL Circuits: Rigorous Mathematical Definitions and Limitations of Classical Theory

The terms hold-in, pull-in (capture), and lock-in ranges are widely used by engineers for the concepts of frequency deviation ranges within which PLL-based circuits can achieve lock under various additional conditions. Usually only non-strict definitions are given for these concepts in engineering literature. After many years of their usage, F. Gardner in the 2nd edition of his well-known work, Phaselock Techniques, wrote “There is no natural way to define exactly any unique lock-in frequency” and “despite its vague reality, lock-in range is a useful concept.” Recently these observations have led to the following advice given in a handbook on synchronization and communications: “We recommend that you check these definitions carefully before using them.” In this survey an attempt is made to discuss and fill some of the gaps identified between mathematical control theory, the theory of dynamical systems and the engineering practice of phase-locked loops. It is shown that, from a mathematical point of view, in some cases the hold-in and pull-in “ranges” may not be the intervals of values but a union of intervals and thus their widely used definitions require clarification. Rigorous mathematical definitions for the hold-in, pull-in, and lock-in ranges are given. An effective solution for the problem on the unique definition of the lock-in frequency, posed by Gardner, is suggested.

Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large

The analysis of the stability and numerical simulation of Costas loop circuits for high-frequency signals is a challenging task. The problem lies in the fact that it is necessary to simultaneously observe very fast time scale of the input signals and slow time scale of phase difference between the input signals. To overcome this difficult situation it is possible, following the approach presented in the classical works of Gardner and Viterbi, to construct a mathematical model of Costas loop, in which only slow time change of signal?s phases and frequencies is considered. Such a construction, in turn, requires the computation of phase detector characteristic, depending on the waveforms of the considered signals. While for the stability analysis of the loop near the locked state (local stability) it is usually sufficient to consider the linear approximation of phase detector characteristic near zero phase error, the global analysis (stability in the large) cannot be accomplished using simple linear models.The present paper is devoted to the rigorous construction of nonlinear dynamical model of classical Costas loop, which allows one to apply numerical simulation and analytical methods (various modifications of absolute stability criteria for systems with cylindrical phase space) for the effective analysis of stability in the large. Here a general approach to the analytical computation of phase detector characteristic of classical Costas loop for periodic non-sinusoidal signal waveforms is suggested. The classical ideas of the loop analysis in the signal?s phase space are developed and rigorously justified. Effective analytical and numerical approaches for the nonlinear analysis of the mathematical model of classical Costas loop in the signal?s phase space are discussed. HighlightsAnalytical computation of PD characteristic of Costas loop is done.The classical ideas of Gardner and Viterbi are developed and rigorously justified.Nonlinear analysis of Costas loop in the signal?s phase space is demonstrated.

Limitations of the classical phase-locked loop analysis

Nonlinear analysis of the classical phase-locked loop (PLL) is a challenging task. In classical engineering literature simplified mathematical models and simulation are widely used for its study. In this work the limitations of classical engineering phase-locked loop analysis are demonstrated, e.g., hidden oscillations, which can not be found by simulation, are discussed. It is shown that the use of simplified mathematical models and the application of simulation may lead to wrong conclusions concerning the operability of PLL-based circuits.

A short survey on nonlinear models of the classic Costas loop: Rigorous derivation and limitations of the classic analysis

Rigorous nonlinear analysis of the physical model of Costas loop-a classic phase-locked loop (PLL) based circuit for carrier recovery, is a challenging task. Thus for its analysis, simplified mathematical models and numerical simulation are widely used. In this work a short survey on nonlinear models of the BPSK Costas loop, used for pre-design and post-design analysis, is presented. Their rigorous derivation and limitations of classic analysis are discussed. It is shown that the use of simplified mathematical models, and the application of non rigorous methods of analysis (e.g., simulation and linearization) may lead to wrong conclusions concerning the performance of the Costas loop physical model.

Nonlinear analysis of classical phase-locked loops in signal's phase space

Abstract Discovery of undesirable hidden oscillations, which cannot be found by the standard simulation, in phase-locked loop (PLL) showed the importance of consideration of nonlinear models and development of rigorous analytical methods for their analysis. In this paper for various signal waveforms, analytical computation of multiplier/mixer phase-detector characteristics is demonstrated, and nonlinear dynamical model of classical analog PLL is derived. Approaches to the rigorous nonlinear analysis of classical analog PLL are discussed.

Limitations of PLL simulation: Hidden oscillations in MatLab and SPICE

Nonlinear analysis of the phase-locked loop (PLL) based circuits is a challenging task, thus in modern engineering literature simplified mathematical models and simulation are widely used for their study. In this work the limitations of numerical approach is discussed and it is shown that, e.g. hidden oscillations may not be found by simulation. Corresponding examples in SPICE and MatLab, which may lead to wrong conclusions concerning the operability of PLL-based circuits, are presented.

Tutorial on dynamic analysis of the Costas loop

Abstract Costas loop is a classical phase-locked loop (PLL) based circuit for carrier recovery and signal demodulation. The PLL is an automatic control system that adjusts the phase of a local signal to match the phase of the input reference signal. This tutorial is devoted to the dynamic analysis of the Costas loop. In particular the acquisition process is analyzed. Acquisition is most conveniently 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 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. Then equations for the above mentioned key parameters are derived. Finally, the lock-in range of the Costas loop for the case where a lead-lag filter is used for the loop filter is analyzed.

Differential equations of Costas loop

The Costas loop was invented in the 1950s [1] and is intended for carrier phase recovery from signals widely applied in communications and control sys tems [2–4]. To describe a mathematical model of the Costas loop in the form of differential equations, it was neces sary to develop asymptotic methods for analyzing high frequency oscillations propagating in nonlinear electronic circuits of special form [5, 6]. In this paper, we generalize the approach described in [5–9]. Specifically, it is proved that the Costas loop in the signal space is asymptotically equivalent to the proposed scheme in the phase frequency space. These results are used to derive differential equations that describe the dynamics of the Costas loop and general ize the results obtained in [9, 10].

Analytical methods for computation of phase-detector characteristics and PLL design

An effective analytical methods for computation of phase detector characteristics are suggested. For high-frequency oscillators new classes of such characteristics are described. Approaches to a rigorous nonlinear analysis of PLL are discussed.