Sinusoidal Phase Detector Research Articles

Study on TPD Phasemeter to Suppress Low-Frequency Amplitude Fluctuation and Improve Fast-Acquiring Range for GW Detection.

A phasemeter as a readout system for the inter-satellite laser interferometer in a space-borne gravitational wave detector requires not only high accuracy but also insensitivity to amplitude fluctuations and a large fast-acquiring range. The traditional sinusoidal characteristic phase detector (SPD) phasemeter has the advantages of a simple structure and easy realization. However, the output of an SPD is coupled to the amplitude of the input signal and has only a limited phase-detection range due to the boundedness of the sinusoidal function. This leads to the performance deterioration of amplitude noise suppression, fast-acquiring range, and loop stability. To overcome the above shortcomings, we propose a phasemeter based on a tangent phase detector (TPD). The characteristics of the SPD and TPD phasemeters are theoretically analyzed, and a fixed-point simulation is further carried out for verification. The simulation results show that the TPD phasemeter tracks the phase information well and, at the same time, suppresses the amplitude fluctuation to the noise floor of 1 μrad/Hz1/2, which meets the requirements of GW detection. In addition, the maximum lockable step frequency of the TPD phasemeter is almost three times larger than the SPD phasemeter, indicating a greater fast-acquiring range.

The Gardner Problem on the Lock-In Range of Second-Order Type 2 Phase-Locked Loops

Phase-locked loops (PLLs) are nonlinear automatic control circuits widely used in telecommunications, computer architecture, gyroscopes, and other applications. One of the key problems of nonlinear analysis of PLL systems has been stated by Floyd M. Gardner as being “to define exactly any unique lock-in frequency.” The lock-in range concept describes the ability of PLLs to reacquire a locked state without cycle slipping and its calculation requires nonlinear analysis. The present work analyzes a second-order type 2 phase-locked loop with a sinusoidal phase detector characteristic. Using the qualitative theory of dynamical systems and classical methods of control theory, we provide stability analysis and suggest analytical lower and upper estimates of the lock-in range based on the exact lock-in range formula for a second-order type 2 PLL with a triangular phase detector characteristic, obtained earlier. Applying phase plane analysis, an asymptotic formula for the lock-in range which refines the existing formula is obtained. The analytical formulas are compared with computer simulation and engineering estimates of the lock-in range. The comparison shows that engineering estimates can lead to cycle slipping in the corresponding PLL model and cannot provide a reliable solution for the Gardner problem, whereas the lower estimate presented in this article guarantees frequency reacquisition without cycle slipping for all parameters, which provides a solution to the Gardner problem.

Exploring the Dynamics of a Third-Order Phase-Locked Loop Model

We study the dynamics and characterize the bifurcation structure of a phase-locked loop (PLL) device modeled by a third-order autonomous differential equation with sinusoidal phase detector. The development of this work was performed using rigorous analysis and numerical experiments. Through theoretical analysis the bifurcation structures related to two fundamental equilibrium points of the system are described. By using extensive numerical experiments we investigate the intricate organization between periodic and chaotic domains in parameter space (named here parameter plane as the PLL model has only two control parameters) and obtain two following remarkable findings: (i) there are self-organized generic stable periodic structures along specific directions in parameter plane, whose periods are defined by a mathematical rule and, (ii) the existence of transient chaos phenomenon responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for some particular control parameter pairs is characterized. Our theoretical and numerical results present an astonishing concordance. We believe that the present study, specially the parameter plane analysis, may have a great importance to experimental studies and general applications involving PLL devices when, for example, one would like to avoid the chaotic regimes.

Computation of the lock-in ranges of phase-locked loops with PI filter

In the present work the lock-in range of PLL-based circuits with proportionally-integrating filter and sinusoidal phase-detector characteristics are studied. Considered circuits have sinusoidal phase detector characteristics. Analytical approach based on the methods of phase plane analysis is applied to estimate the lock-in ranges of the circuits under consideration. Obtained analytical results are compared with simulation results.

An exact formula for the maximum VCO sweep rate of a PLL

In many phase-locked loop (PLL) applications, the natural pull-in mechanism is too slow and unreliable, and it must be accelerated. By adding an externally-generated ramp to its control voltage input, the PLL voltage controlled oscillator (VCO) frequency can be swept towards the input reference frequency in an attempt to speed up the pull-in process. This popular acquisition aid has a significant limitation when it is used in a second-order, Type II PLL. If the applied ramp voltage has a slope magnitude greater than (alternatively, less than) some value Rm, the PLL state can (alternatively, cannot) sweep past the desired lock point, resulting in a phase lock failure (alternatively, success). In general, the maximum sweep rate magnitude Rm can be computed by using a numerical integration-and-search procedure that is described in the PLL literature. A special case exists for a second-order, Type II PLL that incorporates a triangular-characteristic phase detector (with logic-level binary input signals). For this case, it is possible to develop an exact, closed-form expression for Rm, the main result of this paper. For a range of loop parameters most often used in applications, Rm values are computed by using the exact formula, and these are used in two ways. First, they are used to validate the previously-mentioned numerical integration-and-search procedure. Second, they are compared to maximum sweep rate values computed for a PLL that utilizes a sinusoidal phase detector to show that the triangular-phase-detector PLL can be swept significantly faster than the sinusoidal-phase-detector loop.

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

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.

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

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.

An exact formula for the half-plane pull-in range of a PLL

Abstract A second-order phase-lock loop (PLL) that is based on a triangular-characteristic phase detector and imperfect-integrator loop filter is found in many applications where simplicity and economics are major considerations. For many of these applications, digital-logic-compatible reference and VCO signals are used, an exclusive-OR gate implements the phase detector, and the loop filter is constructed from passive components. When designing these loops, the half-plane pull-in range Ω 2 is of interest. Until now, this important loop parameter could only be calculated by using a computer-based technique that numerically integrated the nonlinear differential equation that describes the PLL model. This requirement/limitation is removed here by the development of an exact closed-form formula for Ω 2 , the main contribution of this paper. More generally, the value of Ω 2 is dependent on the PLL phase detector characteristic that is used, be it triangular, sinusoidal, or something else. With regard to the value of Ω 2 produced, a comparison is given of two PLLs, both described by the same linear model so that the comparison is meaningful. The first PLL is based on a triangular-characteristic phase detector; the second loop is based on a sinusoidal phase detector.

Chaos control of third-order phase-locked loops using backstepping nonlinear controller

Previous study showed that a third-order phase-locked loop (PLL) with sinusoidal phase detector characteristics experienced a Hopf bifurcation point as well as chaotic behavior. As a result, this behavior drives the PLL to the out-of-lock (unstable) state. The analysis was based on a modern nonlinear theory such as bifurcation and chaos. The main goal of this paper is to control this chaotic behavior. A nonlinear controller based on the theory of backstepping is designed. The study showed the effectiveness of the designed nonlinear controller in controlling the undesirable unstable behavior and pulling the PLL back to the in-lock state.

Chaos and bifurcation in a third-order phase locked loop

The chaos induced in a new type of phase locked loop (PLL) having a second-order loop filter is investigated. The system under consideration is modeled as a third-order autonomous system with sinusoidal phase detector characteristics. The modern of nonlinear theory such as bifurcation and chaos is applied to a third-order of PLL. A method is developed to quantitatively study the type of bifurcations that occur in this type of PLL’s. The study showed that PLL experiencing a Hopf bifurcation point as well as chaotic behaviour. The method of multiple scales is used to find the normal form near the Hopf bifurcation point. The point is found to be supercritical one.

Numerical treatment of the first-order pll equation with sinusoidal phase detector characteristic

We discuss a powerful method depending on complex continued fractions to obtain and to solve a nonlinear equation for all eigenvalues of the underlying boundary value problem of the first-order phase locked loop equation. Furthermore we give several numerical examples and - for the sake of comparison - we list the first unverified and verified eigenvalues for a relevant signal-to-noise ratio.

Mean cycle slip time of first-order phase-lockedloop withlimiter phase detector

The mean cycle slip time of a first-order phase-locked loop with sinusoidal limiter phase detector is evaluated to account for the dependence of the loop noise spectrum on the phase error. It is shown that neglecting this dependence leads to a considerable underestimation of the mean cycle slip time. Numerical results superior to those for the multiplier type phase detector are reported.

Synchronous behaviour of mutually coupled phase-locked loops in the presence of continuous wave interference

Abstract An analysis is presented concerning the effects of continuous wave interference on the performance of two mutually coupled phase-locked loops with a sinusoidal phase detector characteristic. The parasitic phase error and the parasitic phase modulation introduced into the reconstructed carrier by the interfering signal have been examined for different loop parameters and an input carrier-to-interference ratio. The results show that the interfering signal will only influence the loop in which it occurs and no parasitic phase error will be introduced into the other loop.

The iterated extended Kalman phase detector

Two extended-range phase detectors (PDs), including the tanlock PD, are derived from the iterated extended Kalman filter. Fast converging recursion and concise initialization equations are given. Simulations of a Wiener phase process and a first-order Markov FM process show slightly reduced mean-square error near threshold and faster phase acquisition than a sinusoidal PD.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Phase detector performance in the presence of harmonic distortion

The influence of harmonic distortion on phase errors is investigated. Linear (saw-tooth), triangular and sinusoidal phase detectors are examined. It is shown that these errors are directly related to the harmonic distortion factors of the input signals. The errors are computed for each phase detector.

Exact calculation of phase-locked loop lock-in frequency

Using a novel cycle-slip criterion, two methods of computing the lock-in frequency of phase-locked loops are reported. The lock-in frequency of a second-order loop with sinusoidal phase detector is presented as a function of system parameters, and useful estimates by the pull-out and pull-in frequencies are given.

Phase and Frequency Transfer Between Mutually Synchronized Oscillators

Accurate phase and frequency transfer between mutually synchronized oscillators is a problem of current interest. This paper addresses a system of two mutually synchronized oscillators with inherently sinusoidal phase detectors. The system is studied in the context of nonlinear phase-locked loop theory. This is accomplished by Statistically characterizing the and space and increments of the phase processes generated by the two oscillators. The characterization of these increment processes is obtained in the form of steady-state probability density functions (pdf's). The effects of such disturbances as noise, transmission path delays, and oscillator frequency instabilities on the system are studied through their effects on these pdf's. Gaussian approximations for these pdf's (which simplify the extraction of the effects of the disturbances on the system) are also developed. Additionally, the stability of the time intervals and time interval errors generated by the oscillators are studied in the context of cycle slipping phenomena.

Phase-Locked Loops with Limiter Phase Detectors in the Presence of Noise

A nonlinear mathematical model of phase-locked loops with limiter phase detectors in the presence of noise is presented. The model, which is an extension of the well-known baseband model of loops with sinusoidal phase detectors without limiters, incorporates a modified nonlinear phase detector characteristic, the form of which is changed if the input carrier-to-noise ratio alters, as well as a modified phase noise as an input to the model. Both the modified phase detector characteristic and the spectral density of the modified noise term are calculated specifically for sinusoidal, triangular, and sawtooth types of limiter phase detectors, allowing the application of various methods to determine the loop noise performance. As an example, the phase error variance of a first-order phase-locked loop is calculated, thereby showing a strong dependence on the specific phase detector realization.

Phase Detector Response to Noisy and Noisy Fading Signals

Transfer characteristics of sinusoidal, triangular, and sawtooth-type phase detectors in response to noisy and noisy fading signals have been studied in detail. A new analytical model of the swatooth-type phase detector has been suggested. Detailed experimental results have been supplemented in support of the theoretical findings. Both the theoretical findings and the experimental results clearly indicate that the superiority of a sawtooth-type phase detector over the other two varieties is completely lost in noisy and noisy fading environments.

Sweep-Acquisition Time of Third-Order Generalized Tracking Systems

By using a quasi-stationary approximation a closed form expression is derived for the acquisition time of a third-order swept tracking system, with an arbitrary phase-detector characteristic and with nonideal as well as with ideal integrating filter, optimized for tracking or demodulation in the absence of noise with large initial detuning (relative to the noise bandwidth). For sinusoidal phase detector with ideal or nonideal filters, exact solutions confirm theoretical predictions.