1,028 research outputs found
Solution of the Gardner problem on the lock-in range of phase-locked loop
The lock-in frequency and lock-in range concepts were introduced in 1966 by
Floyd Gardner to describe the frequency differences of phase-locked loop based
circuit for which the loop can acquire lock within one beat, i.e. without cycle
slipping. These concepts became popular among engineering community and were
given in various engineering publications. However rigorous mathematical
explanations these concepts turned out to be a challenging task, thus, in the
2nd edition of Gardner's well-known work, Phaselock Techniques, he wrote that
"despite its vague reality, lock-in range is a useful concept" and posed the
problem "to define exactly any unique lock-in frequency". In this paper an
effective solution for Gardner's problem on the unique definition of the
lock-in frequency and lock-in range is discussed. The lock-in range and lock-in
frequency computation is explained on the example of classical second-order PLL
with lead-lag and active proportional-integral filters. The obtained results
can also be used for the lock-in range computation of such PLL-based circuits
as two-phase PLL, two-phase Costas loop, BPSK Costas loop, and optical Costas
loop, used in intersatellite communication
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 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
Optical Costas loop: pull-in range estimation and hidden oscillations
In this work we consider a mathematical model of the optical Costas loop. The
pull-in range of the model is estimated by analytical and numerical methods.
Difficulties of numerical analysis, related to the existence of so-called
hidden oscillations in the phase space, are discussed.Comment: submitted to IFA
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
Nonlinear analysis of PLL by the harmonic balance method
In this paper we discuss the application of the harmonic balance method for
the global analysis of the classical phase-locked loop (PLL) circuit. The
harmonic balance is non rigorous method, which is widely used %,often without
rigorous justification, for the computation of periodic solutions and the
checking of global stability. The proof of the absence of periodic solutions is
a key step to establish the global stability of PLL and estimate the pull-in
range (which is an interval of the frequency deviations such that any solution
tends to one of the equilibria). The advantages and limitations of the study of
the classical PLL with lead-lag filter using the harmonic balance method is
discussed.Comment: submitted to IFA
Lock-in range of PLL-based circuits with proportionally-integrating filter and sinusoidal phase detector characteristic
In the present work PLL-based circuits with sinusoidal phase detector
characteristic and active proportionally-integrating (PI) filter are
considered. The notion of lock-in range -- an important characteristic of
PLL-based circuits, which corresponds to the synchronization without cycle
slipping, is studied. For the lock-in range a rigorous mathematical definition
is discussed. Numerical and analytical estimates for the lock-in range are
obtained
Lock-in range of classical PLL with impulse signals and proportionally-integrating filter
In the present work the model of PLL with impulse signals and active PI
filter in the signal's phase space is described. For the considered PLL the
lock-in range is computed analytically and obtained result are compared with
numerical simulations.Comment: arXiv admin note: substantial text overlap with arXiv:1603.0840
A short survey on QPSK Costas loop mathematical models
The Costas loop is a modification of the phase-locked loop circuit, which
demodulates data and recovers carrier from the input signal. The Costas loop is
essentially a nonlinear control system and its nonlinear analysis is a
challenging task. Thus, simplified mathematical models and their numerical
simulation are widely used for its analysis. At the same time for phase-locked
loop circuits there are known various examples where the results of such
simplified analysis are differ substantially from the real behavior of the
circuit. In this survey the corresponding problems are demonstrated and
discussed for the QPSK Costas loop.Comment: submitted to IFA
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 dynamical models and the application of simulation
may lead to wrong conclusions concerning the operability of PLL-based circuits
Hidden and self-excited attractors in Chua circuit: SPICE simulation and synchronization
Nowadays various chaotic secure communication systems based on
synchronization of chaotic circuits are widely studied. To achieve
synchronization, the control signal proportional to the difference between the
circuits signals, adjust the state of one circuit. In this paper the
synchronization of two Chua circuits is simulated in SPICE. It is shown that
the choice of control signal is be not straightforward, especially in the case
of multistability and hidden attractors
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