Lock Range Research Articles

Computation of lock-in range for classic PLL with lead-lag filter and impulse signals

For a classic PLL with square waveform signals and lead-lag filter for all possible parameters lock-in range is computed and corresponding diagrams are given.

An On-Chip Measurement of PLL Transfer Function and Lock Range through Fully Digital Interface

An On-Chip Measurement of PLL Transfer Function and Lock Range through Fully Digital Interface

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.

A Wide Locking Range and Low Power Divide-by-2/3 LC Injection-Locked Frequency Divider

A direct divide-by-2/3 LC injection-locked frequency divider (ILFD) is presented in this paper. The proposed ILFD circuit is based on a CMOS LC tank oscillator coupled with an injection NMOS transistor in series with an inductor. Together with body-biased technique and current-reused topology, the locking range of the ILFD is improved and the power consumption is reduced. The circuit is implemented in a 0.18[Formula: see text][Formula: see text]m standard RF CMOS process. At the incident power of 0[Formula: see text]dBm, the measured locking range of the divide-by-2 and divide-by-3 modes are from 5.37[Formula: see text]GHz to 7.68 (8.07[Formula: see text]GHz to 11.4[Formula: see text]GHz) GHz, and the core circuit without buffers consumes 3[Formula: see text]mW at the supply voltage of 1.5[Formula: see text]V. The chip only occupies [Formula: see text] without the pads and the buffers.

Design and Analysis of Low-Power High-Frequency Robust Sub-Harmonic Injection-Locked Clock Multipliers

A low-jitter, low-power LC-based injection-locked clock multiplier (ILCM) with a digital frequency-tracking loop (FTL) is presented. Based on a pulse gating technique, the proposed FTL continuously tunes the oscillator’s free-running frequency to ensure robust operation across PVT variations. The FTL resolves the race condition existing in injection-locked PLLs by decoupling frequency tuning from the injection path, such that the phase-locking condition is only determined by the injection path. This paper also introduces an accurate theoretical large-signal analysis for phase domain response (PDR) of injection-locked oscillators (ILOs). The proposed PDR analysis captures the asymmetric nature of ILO’s lock-in range, and the impact of frequency error on injection strength and phase noise performance. The proposed architecture and analysis are demonstrated by a prototype fabricated in 65 nm CMOS process with active area of $0.25\;\text{mm}^2$ . The prototype ILCM generates output clock in the range of 6.75–8.25 GHz by multiplying the reference clock by 64. It achieves superior integrated jitter performance of $190\;\text{fs}_{\text{rms}}$ , while consuming 2.25 mW power. This translates to an excellent figure-of-merit (FoM) of $-251\;\text{dB}$ , which is the best reported high-frequency clock multiplier.

Measurements and improved model of vortex-induced vibration for an elongated rectangular cylinder

This paper deals with vortex-induced vibrations of a rectangular 4:1 cylinder with a twofold purpose. First, it provides experimental results as a benchmark for checking the predictions of mathematical models and computational techniques. In particular, a wide wind tunnel campaign on a sectional model in smooth flow yielded the oscillation amplitudes in the lock-in range for several Scruton numbers. The dominant frequency of the velocity fluctuations in the wake of the oscillating body was measured. The second purpose of the paper is the improvement of a current approach for assessing vortex-induced vibrations of bridge decks. For this reason, prior to starting the analysis an overview of the existing mathematical models is reported. Afterwards, it is shown through wind tunnel tests that the single-degree-of-freedom model developed by Scanlan at the beginning of the 1980s is unsuitable to predict the VIV response of bridge decks for Scruton numbers different from that at which the model parameters were identified. Finally, an improved version of the model is proposed, allowing to approximate accurately the limit-cycle amplitudes for the entire range of Scruton numbers tested. The identification of the aeroelastic parameters of the improved version of the model requires only three decay-to-resonance tests for three different Scruton numbers. From the engineering point of view, this represents an important advantage, especially when the mass and the damping of the structure are uncertain or dampers have to be designed, as it makes the assessment of the sensitivity to vortex-induced vibration of bridge deck cross sections less time-consuming and expensive. In the future, it will be necessary to verify that the functional form postulated for the aeroelastic parameters of the model is valid not only for elongated rectangular cylinders but also for bridge deck cross sections.

An Efficient Method to Analyze Lock Range in Ring Oscillators With Multiple Injections

In this brief, a simple method is proposed to analyze the lock range (LR) of ring oscillators with multiple injection signals by utilizing the generalized Adler equation. An exact formula is introduced for the LR in the case of a single injection. The proposed method is employed to study the asymmetry in high and low LRs. To describe the dependence of the oscillator LR to injection phases in the case of multiple injections, a simple closed-form formula and an empirical formula are derived in weak injection and strong injection, respectively. All of the proposed mathematical findings comprise the results of recent works that have been done in this area and are verified by circuit simulation.

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.

Design and Tuning of Coupled-LC mm-Wave Subharmonically Injection-Locked Oscillators

This paper analyzes and demonstrates the use of coupled-LC tanks to enlarge the locking range (LR) of millimeter-wave (mm-wave) subharmonically injection-locked oscillators. Design guidelines are derived from a simplified analysis. Different techniques are proposed to tune the coupled-LC tank. A mm-wave subharmonically injection-locked quadrature voltage-controlled oscillator in 40-nm CMOS verifies the proposed approach. The LR is larger than 2 GHz over a 55–63-GHz tuning range. An on-chip envelope detector facilitates the tuning of the coupled-LC tank. The phase noise and the quadrature-phase imbalance are uniform over almost the whole LR.

The influence of higher harmonic flow forces on the response of a curved circular cylinder undergoing vortex-induced vibration

Vortex-induced vibration (VIV) of a curved circular cylinder (a quarter of a ring, with no extension added to either end) free to oscillate in the crossflow direction was studied experimentally. Both the concave and the convex orientations (with respect to the oncoming flow direction) were considered. As expected, the amplitude of oscillations in both configurations was decreased compared to a vertical cylinder with the same mass ratio. Flow visualizations showed that the vortices were shed in parallel to the curved cylinder, when the cylinder was free to oscillate. The sudden jump in the phase difference between the flow forces and the cylinder displacement observed in the VIV of vertical cylinders was not observed in the curved cylinders. Higher harmonic force components at frequencies twice and three times the frequency of oscillations were observed in flow forces acting on the vertical cylinder, as well as the curved cylinder. Asymmetry in the wake was responsible for the 2nd harmonic force component and the relative velocity of the structure with respect to the oncoming flow was responsible for the 3rd harmonic force component. The lock-in occurred over the same range of reduced velocities for the curved cylinder in the convex orientation as for a vertical cylinder, but it was extended to higher reduced velocities for a curved cylinder in the concave orientation. Higher harmonic force components were found to be responsible for the extended lock-in range in the concave orientation. Within this range, the higher harmonic forces were even larger than the first harmonic force and the structure was being excited mainly by these higher harmonic forces.

An experimental investigation of vortex-induced vibration of a rotating circular cylinder in the crossflow direction

Vortex-induced vibration of a flexibly-mounted circular cylinder free to oscillate in the crossflow direction with imposed rotation around its axis was studied experimentally. The rotation rate, α, defined as the ratio of the surface velocity and free stream velocity, was varied from 0 to 2.6 in small steps. The amplitudes and frequencies of oscillations as well as the flow forces were measured in a Reynolds number range of Re = 350 -1000. The maximum amplitude of oscillations was limited to values less than a diameter of the cylinder at high rotation rates. Also, the lock-in range became narrower at higher rotation rates and finally the oscillations ceased beyond α = 2.4. Vortex shedding pattern was found to be 2S (two single vortices shed per cycle of oscillations) for rotation rates up to α = 1.4 and transitioned toward an asymmetric P shedding (one pair of vortices shed in a cycle of oscillations) for rotation rates within the range of 1.4 ≤ α ≤ 1.8. Vortex shedding was found to persist up to higher rotation rates than those observed for a non-oscillating cylinder. The phase difference between the flow forces and displacement of the cylinder in the crossflow direction was influenced as the rotation rate was increased: At high reduced velocities, the phase difference decreased from 180° for a non-rotating cylinder to values close to 90° for a rotating cylinder at large rotation rates. Different shedding patterns resulted in flow forces with different frequencies. In the crossflow direction, the dominant frequency of flow forces was found to be close to the system’s natural frequency for all the rotation rates tested with either 2S or P vortex shedding pattern. In the inline direction, however, the change from 2S to P shedding at high rotation rates resulted in a shift of the ratio of the dominant frequency of the inline flow forces to the natural frequency of the system from 2:1 to 1:1.

FIR digital filter-based ZCDPLL for carrier recovery

The objective of this work is to analyse the performance of the newly proposed two-tap FIR digital filter-based first-order zero-crossing digital phase-locked loop (ZCDPLL) in the absence or presence of additive white Gaussian noise (AWGN). The introduction of the two-tap FIR digital filter widens the lock range of a ZCDPLL and improves the loop’s operation in the presence of AWGN. The FIR digital filter tap coefficients affect the loop convergence behaviour and appropriate selection of those gains should be taken into consideration. The new proposed loop has wider locking range and faster acquisition time and reduces the phase error variations in the presence of noise.

Study of injection‐locked non‐harmonic oscillators using Volterra series

This study presents a Volterra series approach to analyse injection-locked non-harmonic oscillators. We show that by depicting the voltage transfer characteristics of comparators using hyperbolic tangent functions, non-harmonic oscillators can be analysed analytically using a set of Volterra circuits that are linear, have the same topology and element values but different inputs. We further show that the larger lock range of non-harmonic oscillators as compared with that of their harmonic counterparts is because of the harsher non-linear characteristics of these oscillators and the lower-order attenuation of the high-order frequency components of the oscillators. The reduced non-linear characteristics of ring oscillators because of the absence of positive feedback also gives rise to a smaller lock range as compared with relaxation oscillators. These theoretical findings are validated using both the simulation results of relaxation oscillators and ring oscillators designed in IBM 130 nm complementary metal oxide semiconductor technology and the measurement results of ring oscillators implemented using commercial ICs.

A modified non-linear model for high mass ratio square cylinder

An improved analytical method for the modelling of vortex induced vibrations (VIV) of a square section cylinder is reported. The method is based on a typical structural oscillator coupled to a non-linear wake (Van der Pol) oscillator. Improvements are introduced to the pre-existing analytical methods sorted out from the literature. Dissipation term in the structural oscillator model is modified by the addition of a non-linear damping term. Definition of the wake oscillator is updated simultaneously to include stream-wise force component exerted on the structure. Results show that our modified analytical method correctly predicts the location of peak lock-in amplitude on the amplitude vs reduced-velocity map as observed in recent experiments. The range of lock-in is also reasonably well predicted. The modified method shows that the influence of non-linear structural damping and stream-wise force components cannot be neglected while modelling high mass ratio systems.

A Silicon-Based 0.3 THz Frequency Synthesizer With Wide Locking Range

A 300 GHz frequency synthesizer incorporating a triple-push VCO with Colpitts-based active varactor (CAV) and a divider with three-phase injection is introduced. The CAV provides frequency tunability, enhances harmonic power, and buffers/injects the VCO fundamental signal from/to the divider. The locking range of the divider is vastly improved due to the fact that the three-phase injection introduces larger allowable phase change and injection power into the divider loop. Implemented in 90 nm SiGe BiCMOS, the synthesizer achieves a phase-noise of -77.8 dBc/Hz (-82.5 dBc/Hz) at 100 kHz (1 MHz) offset with a crystal reference, and an overall locking range of 280.32-303.36 GHz (7.9%).

Streamwise vortex-induced vibrations of cylinders with one and two degrees of freedom

AbstractMeasurements are presented of the structural response and wake of a two-degree-of-freedom (2-DOF) pivoted cylinder undergoing streamwise vortex-induced vibrations (VIV), which were carried out using particle-image velocimetry (PIV). The results are compared with those of previous studies performed in the same experimental facility examining a cylinder free to move only in the streamwise direction (1-DOF). The aim of this study is to examine to what extent the results of previous work on streamwise-only VIV can be extrapolated to the more practical, multi-DOF case. The response regimes measured for the 1- and 2-DOF cases are similar, containing two response branches separated by a low-amplitude region. The first branch is characterised by negligible transverse motion and the appearance of both alternate and symmetric vortex shedding. The two wake modes compete in an unsteady manner; however, the competition does not appear to have a significant effect on either the streamwise or transverse motion. Comparison of the phase-averaged vorticity fields acquired in the second response branch also indicates that the additional DOF does not alter the vortex-shedding process. However, the additional DOF affects the cylinder-wake system in other ways; for the 1-DOF case the second branch can appear in three different forms (each associated with a different wake mode), while for the 2-DOF case the second branch only exists in one form, and does not exhibit hysteresis. The cylinder follows a figure-of-eight trajectory throughout the lock-in range. The phase angle between the streamwise and transverse motion decreases linearly with reduced velocity. This work highlights the similarities and differences between the fluid–structure interaction and wake dynamics associated with 1- and 2-DOF cylinders throughout the streamwise response regime, which has not received attention to date.

The influence of taper ratio on vortex-induced vibration of tapered cylinders in the crossflow direction

A series of experiments were conducted to study vortex-induced vibration of tapered cylinders free to oscillate in the crossflow direction. These experiments were performed in a re-circulating water tunnel with an open-surface test section for a Reynolds number range of Re=370–2300. Three linearly tapered cylinders with taper ratios of 29:1, 17:1 and 10:1 and a cylinder with a nonlinear taper were studied. For all tests, the mass ratio was kept constant. The maximum amplitude of the crossflow oscillations stayed almost constant for all taper ratios, at a value close to the maximum amplitude of a uniform cylinder with the same mass ratio. As the taper ratio was decreased, the lock-in range started at larger reduced velocities and sustained through higher reduced velocities. Tests were conducted for cylinders placed vertically in the test-section of the water tunnel, both with their diameter increasing downward and upward to show that the results were independent from the cylinder׳s orientation. Hybrid shedding modes (both 2S and 2P shedding over the cylinder׳s length) were observed for all the linearly tapered cases for reduced velocities up to the value corresponding to the peak amplitude of oscillations. For higher reduced velocities, only 2P shedding was observed along the length.

Flow past two tandem square cylinders vibrating transversely in phase

Numerical investigations have been carried out to study the wake characteristics of flow past two tandem square cylinders vibrating in phase. Both the cylinders vibrate in a transverse direction, i.e., perpendicular to the incoming flow with the same frequency and amplitude. The frequency of vibration of the cylinders and the inter-cylinder spacing are varied for fixed values of the Reynolds number (Re = 100) and the amplitude ratio ( = 0.4). The synchronous or lock-in regime for the oscillatory wake of the vibrating cylinders has been identified by varying the frequency of the vibration from = 0.4 to 1.6 ( being the frequency of vortex shedding behind a stationary square cylinder). The characteristics of lift and drag and the mechanism of vortex shedding are studied by varying the excitation frequency within the lock-in range for each value of inter-cylinder spacing. The complex interaction of flow between the cylinders gives rise to a variety of characteristically different shedding patterns in their wake. For values of inter-cylinder spacing equal to 2D and 3D, periodic, as well as quasi-periodic, lock-in behaviors are observed in the synchronous range.

Lock-in study of two side-by-side cylinders of different diameters in close proximity in steady flow

Lock-in of the vortex-induced vibration of two side-by-side circular cylinders of different diameters (diameter ratio d/D=0.1) is investigated numerically. The cylinders are located in close proximity and free to oscillate in the cross-flow direction. The initial gap between the two cylinders is set the same as the small cylinder diameter (d). The mass ratios of both cylinders (m⁎) are fixed to be 5 and the damping ratios are small enough to be negligible. Simulations are first carried out for two cases where the large-to-small-cylinder natural frequency ratio is 1. Case 1 is focused on the lock-in of the large cylinder and Case 2 is focused on the lock-in of the small cylinder, which is far narrower than that of the large cylinder. Then simulations are carried out at a natural frequency ratio (small-to-large-cylinder) of 0.1, where both cylinders are expected to lock on to their own natural frequencies (referred to be Case 3). The interference between the two cylinders under these conditions is investigated in detail. The widening of the lock-in range of the reduced velocities for the large cylinder in Case 1, the beating behavior of the small cylinder in its lock-in range in Case 2 and the dual lock-in behavior of the small cylinder during the simultaneous lock-in of both cylinders in Case 3 are some of the key findings of this study.

A New Transmission Gate Cascode Current Mirror Charge Pump for Fast Locking Low Noise PLL

Charge Pump in a phase locked loop (PLL) generates non-ideal effects such as current mismatches at the output node and switching errors at the pull up and pull down networks. This work presents a novel transmission gate cascode current mirror charge pump circuit. The switches incorporated in this work are Transmission Gates which help to reduce various switching errors, and only one supply independent reference current source is used to have a minimum current mismatch. The performance analysis carried out in the Cadence design environment, and it is observed that the loop locks in 25 ns which is 50 % faster than the conventional charge pump. The control voltage absolutely has no ripple in it after locking which reduces the reference spur further. It could be achieved because the current mismatch is only about 7 %. This PLL operates at 2.5 GHz having a wide lock range of 0.5---2.8 GHz where average power consumption is 1.74 mW. Due to the use of cascade current mirror circuits, the output voltage swing that can be obtained is 1.79 V.