Search results for: pitching oscillation

Authors: M. R. Soltani, M. Seddighi, M. Mahmoudi

Abstract:

Extensive wind tunnel tests have been conducted to investigate the unsteady flow field over and behind a 2D model of a 660 kW wind turbine blade section in pitching motion. The surface pressure and wake dynamic pressure variation at a distance of 1.5 chord length from trailing edge were measured by pressure transducers during several oscillating cycles at 3 reduced frequencies and oscillating amplitudes. Moreover, form drag and linear momentum deficit are extracted and compared at various conditions. The results show that the wake velocity field and surface pressure of the model have similar behavior before and after the airfoil beyond the static stall angle of attack. In addition, the effects of reduced frequency and oscillation amplitudes are discussed.

Keywords: Pitching motion, form drag, Profile drag, windturbine.

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Authors: Sadeghi H., Mani M., Ardakani M. A.

Abstract:

The unsteady wake of an EPPLER 361 airfoil in pitching motion has been investigated in a subsonic wind tunnel by hot-wire anemometry. The airfoil was given the pitching motion about the one-quarter chord axis at reduced frequency of 0182. Streamwise mean velocity profiles (wake profiles) were investigated at several vertically aligned points behind the airfoil at one-quarter chord downstream distance from trailing edge. Oscillation amplitude and mean angle of attack were varied to determine the effects on wake profiles. When the maximum dynamic angle of attack was below the static stall angle of attack, weak effects on wake were found by increasing oscillation amplitude and mean angle of attack. But, for higher angles of attack strong unsteady effects were appeared on the wake.

Keywords: Unsteady wake, amplitude, mean angle, EPPLER 361 airfoil.

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Authors: Mahbod Seyednia, Shidvash Vakilipour, Mehran Masdari

Abstract:

Computations for two-dimensional flow past a stationary and harmonically pitching wind turbine airfoil at a moderate value of Reynolds number (400000) are carried out by progressively increasing the angle of attack for stationary airfoil and at fixed pitching frequencies for rotary one. The incompressible Navier-Stokes equations in conjunction with Unsteady Reynolds Average Navier-Stokes (URANS) equations for turbulence modeling are solved by OpenFOAM package to investigate the aerodynamic phenomena occurred at stationary and pitching conditions on a NACA 6-series wind turbine airfoil. The aim of this study is to enhance the accuracy of numerical simulation in predicting the aerodynamic behavior of an oscillating airfoil in OpenFOAM. Hence, for turbulence modelling, k-ω-SST with low-Reynolds correction is employed to capture the unsteady phenomena occurred in stationary and oscillating motion of the airfoil. Using aerodynamic and pressure coefficients along with flow patterns, the unsteady aerodynamics at pre-, near-, and post-static stall regions are analyzed in harmonically pitching airfoil, and the results are validated with the corresponding experimental data possessed by the authors. The results indicate that implementing the mentioned turbulence model leads to accurate prediction of the angle of static stall for stationary airfoil and flow separation, dynamic stall phenomenon, and reattachment of the flow on the surface of airfoil for pitching one. Due to the geometry of the studied 6-series airfoil, the vortex on the upper surface of the airfoil during upstrokes is formed at the trailing edge. Therefore, the pattern flow obtained by our numerical simulations represents the formation and change of the trailing-edge vortex at near- and post-stall regions where this process determines the dynamic stall phenomenon.

Keywords: CFD, Moderate Reynolds number, OpenFOAM, pitching oscillation, unsteady aerodynamics, wind turbine.

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Authors: Hadi Zarafshani, Shidvash Vakilipour, Shahin Teimori, Sara Barati

Abstract:

In the present study, the aerodynamic performance of a rigid two-dimensional pitching bio-inspired corrugate airfoil was numerically investigated at Reynolds number of 14000. The Open Field Operations And Manipulations (OpenFOAM) computational fluid dynamic tool is used to solve flow governing equations numerically. The k-ω SST turbulence model with low Reynolds correction (k-ω SST LRC) and the pimpleDyMFOAM solver are utilized to simulate the flow field around pitching bio-airfoil. The lift and drag coefficients of the airfoil are calculated at reduced frequencies k=1.24-4.96 and the angular amplitude of A=5^°-20^°. Results show that in a fixed reduced frequency, the absolute value of the sectional lift and drag coefficients increase with increasing pitching amplitude. In a fixed angular amplitude, the absolute value of the lift and drag coefficients increase as the pitching reduced frequency increases.

Keywords: Bio-inspired pitching airfoils, OpenFOAM, low Reynolds k-ω SST model, lift and drag coefficients.

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Authors: M. Tsebia, H. Bentarzi

Abstract:

In the inter-connected power systems, a phenomenon called inter-area oscillation may be caused by several defects. In this paper, a study of the Maghreb countries inter-area power networks oscillation has been investigated. The inter-area oscillation monitoring can be enhanced by integrating Phasor Measurement Unit (PMU) technology installed in different places. The data provided by PMU and recorded by PDC will be used for the monitoring, analysis, and control purposes. The proposed approach has been validated by simulation using MATLAB/Simulink.

Keywords: Inter-area oscillation, Maghrebian power system, Simulink, PMU.

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Authors: Qasem M. Al-Mdallal

Abstract:

This paper presents results obtained from the numerical solution for the flow past an oscillating circular cylinder at Reynolds number of 200. The frequency of oscillation was fixed to the vortex shedding frequency from a fixed cylinder, f0, while the amplitudes of oscillations were varied from to 1.1a, where a represents the radius of the cylinder. The response of the flow through the fluid forces acting on the surface of the cylinder are investigated. The lock-on phenomenon is captured at low oscillation amplitudes.

Keywords: Lock-on; streamwise oscillation; transverse oscillation; fluid forces, combined motion

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Authors: Vasilij Djackov, Tomas Zapnickas, Evgenii Iamshchikov, Lukas Norkevicius, Rima Mickeviciene, Larisa Vasiljeva

Abstract:

In this paper, the resistance and pitching values of the test of an electric ferry are presented. The research was carried out in the open flow channel of Klaipėda University with a multi-axis dynamometer. The received model resistance values were recalculated to the real vessel and the preliminary chosen propulsion unit power was compared. After analyzing the results of the pitching of the model, it was concluded that the shape of the hull needs to be further improved, taking into account the possible uneven weight distribution at the ends of the ferry. Further investigation of the hull of the electric ferry is recommended, including experiments with various water depths and activation of propulsion units.

Keywords: Electrical ferry, model tests, open flow channel, pitching, resistance.

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Authors: Xie Kai, Laith K. Abbas, Chen Dongyang, Yang Fufeng, Rui Xiaoting

Abstract:

Effect of plunging motion on the pitch oscillating NACA0012 airfoil is investigated using computational fluid dynamics (CFD). A simulation model based on overset grid technology and k - ω shear stress transport (SST) turbulence model is established, and the numerical simulation results are compared with available experimental data and other simulations. Two cases of phase angle φ = 0, μ which represents the phase difference between the pitching and plunging motions of an airfoil are performed. Airfoil vortex generation, moving, and shedding are discussed in detail. Good agreements have been achieved with the available literature. The upward plunging motion made the equivalent angle of attack less than the actual one during pitching analysis. It is observed that the formation of the stall vortex is suppressed, resulting in a decrease in the lift coefficient and a delay of the stall angle. However, the downward plunging motion made the equivalent angle of attack higher the actual one.

Keywords: Dynamic stall, pitching-plunging, computational fluid dynamics, helicopter blade rotor, airfoil.

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Authors: Da-Xue Chen, Guang-Hui Liu

Abstract:

In this paper, we study the oscillation of a class of second-order nonlinear neutral damped variable delay dynamic equations on time scales. By using a generalized Riccati transformation technique, we obtain some sufficient conditions for the oscillation of the equations. The results of this paper improve and extend some known results. We also illustrate our main results with some examples.

Keywords: Oscillation theorem, second-order nonlinear neutral dynamic equation, variable delay, damping, Riccati transformation.

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Authors: Debojyoti Mitra

Abstract:

The present study concentrates on solving the along wind oscillation problem of a tall square building from first principles and across wind oscillation problem of the same from empirical relations obtained by experiments. The criterion for human comfort at the worst condition at the top floor of the building is being considered and a limiting value of height of a building for a given cross section is predicted. Numerical integrations are carried out as and when required. The results show severeness of across wind oscillations in comparison to along wind oscillation. The comfort criterion is combined with across wind oscillation results to determine the maximum allowable height of a building for a given square cross-section.

Keywords: Tall Building, Along-wind Response, Across-wind Response, Human Comfort.

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Authors: Yahya Naderi, Tohid Rahimi, Babak Yousefi, Seyed Hossein Hosseini

Abstract:

Today’s modern interconnected power system is highly complex in nature. In this, one of the most important requirements during the operation of the electric power system is the reliability and security. Power and frequency oscillation damping mechanism improve the reliability. Because of power system stabilizer (PSS) low speed response against of major fault such as three phase short circuit, FACTs devise that can control the network condition in very fast time, are becoming popular. But FACTs capability can be seen in a major fault present when nonlinear models of FACTs devise and power system equipment are applied. To realize this aim, the model of multi-machine power system with FACTs controller is developed in MATLAB/SIMULINK using Sim Power System (SPS) blockiest. Among the FACTs device, Static synchronous series compensator (SSSC) due to high speed changes its reactance characteristic inductive to capacitive, is effective power flow controller. Tuning process of controller parameter can be performed using different method. But Genetic Algorithm (GA) ability tends to use it in controller parameter tuning process. In this paper firstly POD controller is used to power oscillation damping. But in this station, frequency oscillation dos not has proper damping situation. So FOD controller that is tuned using GA is using that cause to damp out frequency oscillation properly and power oscillation damping has suitable situation.

Keywords: Power oscillation damping (POD), frequency oscillation damping (FOD), Static synchronous series compensator (SSSC), Genetic Algorithm (GA).

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Authors: Mohammed F. Alhaddad

Abstract:

Vortex Induced Vibrations (VIV) is a phenomenon that occurs as a result of a flow passing by a bluff body. The aim of this paper is to identify factors for maximizing oscillation amplitude generated by VIV in order to enhance the energy harnessed through this method. The experimental study in this paper will examine the effect of oscillating cylinder diameter, surface roughness, the location of surface roughness with respect to the centreline of the oscillating cylinder and the velocity on the oscillation amplitude of the used module.

Keywords: Energy, renewable, electrostatic, vibration, vortex.

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Authors: Da-Xue Chen, Guang-Hui Liu

Abstract:

In this paper, we establish several oscillation criteria for the nonlinear second-order damped delay dynamic equation r(t)|xΔ(t)|β-1xΔ(t)Δ + p(t)|xΔσ(t)|β-1xΔσ(t) + q(t)f(x(τ (t))) = 0 on an arbitrary time scale T, where β > 0 is a constant. Our results generalize and improve some known results in which β > 0 is a quotient of odd positive integers. Some examples are given to illustrate our main results.

Keywords: Oscillation, damped delay dynamic equation, time scale.

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Authors: N. Popovich, P. Yan

Abstract:

In this paper a controller for the pitch angle of an aircraft regarding to the elevator deflection angle is designed. The way how the elevator angle affects pitching motion of the aircraft is pointed out, as well as, how a pitch controller can be applied for the aircraft to reach certain pitch angle. In this digital optimal system, the elevator deflection angle and pitching angle of the plane are considered to be input and output respectively. A single input single output (SISO) system is presented. A digital pitch aircraft control is demonstrated. A simulation for the whole system has been performed. The optimal control weighting vectors, Q and R have been determined.

Keywords: Aircraft, control, digital, optimal, Q and Rmatrices.

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Authors: Pavlo Selyshchev

Abstract:

A theoretical approach to radiation damage evolution is developed. Stable temporal behavior taking place in solids under irradiation are examined as phenomena of self-organization in nonequilibrium systems. Experimental effects of temporal self-organization in solids under irradiation are reviewed. Their essential common properties and features are highlighted and analyzed. Dynamical model to describe development of self-oscillation of density of point defects under stationary irradiation is proposed. The emphasis is the nonlinear couplings between rate of annealing and density of defects that determine the kind and parameters of an arising self-oscillation. The field of parameters (defect generation rate and environment temperature) at which self-oscillations develop is found. Bifurcation curve and self-oscillation period near it is obtained.

Keywords: Irradiation, Point Defects, Solids, Temporal Selforganization.

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Authors: M Nayeripour, H. Khorsand, A. Roosta, T. Niknam, E. Azad

Abstract:

Series compensators have been used for many years, to increase the stability and load ability of transmission line. They compensate retarded or advanced volt drop of transmission lines by placing advanced or retarded voltage in series with them to compensate the effective reactance, which cause to increase load ability of transmission lines. In this paper, two method of fuzzy controller, based on power reference tracking and impedance reference tracking have been developed on TCSC controller in order to increase load ability and improving power oscillation damping of system. In these methods, fire angle of thyristors are determined directly through the special Rule-bases with the error and change of error as the inputs. The simulation results of two area four- machines power system show the good performance of power oscillation damping in system. Comparison of this method with classical PI controller shows the increasing speed of system response in power oscillation damping.

Keywords: TCSC, Two area network, Fuzzy controller, Power oscillation damping.

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Authors: Muhammad Amjad Sohail, Rizwan Ullah

Abstract:

This research paper presents the CFD analysis of oscillating airfoil during pitch cycle. Unsteady subsonic flow is simulated for pitching airfoil at Mach number 0.283 and Reynolds number 3.45 millions. Turbulent effects are also considered for this study by using K-ω SST turbulent model. Two-dimensional unsteady compressible Navier-Stokes code including two-equation turbulence model and PISO pressure velocity coupling is used. Pressure based implicit solver with first order implicit unsteady formulation is used. The simulated pitch cycle results are compared with the available experimental data. The results have a good agreement with the experimental data. Aerodynamic characteristics during pitch cycles have been studied and validated.

Keywords: Angle of attack, Centre of pressure, subsonic flow, pitching moment coefficient, turbulence mode

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Authors: Gye-rok Jeon, Jae-hee Jung, In-cheol Kim, Ah-young Jeon, Sang-hwa Yoon, Jung-man Son, Jae-hyung Kim, Soo-young Ye, Jung-hoon Ro, Dong-hyun Kim, Chul-han Kim

Abstract:

A analysis on the conventional the blood pressure estimation method using an oscillometric sphygmomanometer was performed through a computer simulation using an arterial pressure-volume (APV) model. Traditionally, the maximum amplitude algorithm (MAP) was applied on the oscillation waveforms of the APV model to obtain the mean arterial pressure and the characteristic ratio. The estimation of mean arterial pressure and characteristic ratio was significantly affected with the shape of the blood pressure waveforms and the cutoff frequency of high-pass filter (HPL) circuitry. Experimental errors are due to these effects when estimating blood pressure. To find out an algorithm independent from the influence of waveform shapes and parameters of HPL, the volume oscillation of the APV model and the phase shift of the oscillation with fast fourier transform (FFT) were testified while increasing the cuff pressure from 1 mmHg to 200 mmHg (1 mmHg per second). The phase shift between the ranges of volume oscillation was then only observed between the systolic and the diastolic blood pressures. The same results were also obtained from the simulations performed on two different the arterial blood pressure waveforms and one hyperthermia waveform.

Keywords: Arterial blood pressure, oscillometric method

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Authors: Mohamad Reza. Mohaghegh, Majid. Malek Jafarian

Abstract:

This research proposes an algorithm for the simulation of time-periodic unsteady problems via the solution unsteady Euler and Navier-Stokes equations. This algorithm which is called Time Spectral method uses a Fourier representation in time and hence solve for the periodic state directly without resolving transients (which consume most of the resources in a time-accurate scheme). Mathematical tools used here are discrete Fourier transformations. It has shown tremendous potential for reducing the computational cost compared to conventional time-accurate methods, by enforcing periodicity and using Fourier representation in time, leading to spectral accuracy. The accuracy and efficiency of this technique is verified by Euler and Navier-Stokes calculations for pitching airfoils. Because of flow turbulence nature, Baldwin-Lomax turbulence model has been used at viscous flow analysis. The results presented by the Time Spectral method are compared with experimental data. It has shown tremendous potential for reducing the computational cost compared to the conventional time-accurate methods, by enforcing periodicity and using Fourier representation in time, leading to spectral accuracy, because results verify the small number of time intervals per pitching cycle required to capture the flow physics.

Keywords: Time Spectral Method, Time-periodic unsteadyflow, Discrete Fourier transform, Pitching airfoil, Turbulence flow

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Authors: S. Summart, C. Saetiaw, T. Thosdeekoraphat, C. Thongsopa

Abstract:

This article presents new current-mode oscillator circuits using CDTAs which is designed from block diagram. The proposed circuits consist of two CDTAs and two grounded capacitors. The condition of oscillation and the frequency of oscillation can be adjusted by electronic method. The circuits have high output impedance and use only grounded capacitors without any external resistor which is very appropriate to future development into an integrated circuit. The results of PSPICE simulation program are corresponding to the theoretical analysis.

Keywords: Current-mode, Quadrature Oscillator, Block Diagram, CDTA.

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Authors: Kyoungwoo Park, Byeong-Sam Kim

Abstract:

Unmanned aerial vehicles (UAVs) performing their operations for a long time have been attracting much attention in military and civil aviation industries for the past decade. The applicable field of UAV is changing from the military purpose only to the civil one. Because of their low operation cost, high reliability and the necessity of various application areas, numerous development programs have been initiated around the world. To obtain the optimal solutions of the design variable (i.e., sectional airfoil profile, wing taper ratio and sweep) for high performance of UAVs, both the lift and lift-to-drag ratio are maximized whereas the pitching moment should be minimized, simultaneously. It is found that the lift force and lift-to-drag ratio are linearly dependent and a unique and dominant solution are existed. However, a trade-off phenomenon is observed between the lift-to-drag ratio and pitching moment. As the result of optimization, sixty-five (65) non-dominated Pareto individuals at the cutting edge of design spaces that are decided by airfoil shapes can be obtained.

Keywords: Unmanned aerial vehicle (UAV), Airfoil, CFD, Shape optimization, Genetic Algorithm.

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Authors: I.Banerjee, A.K.Mahendra, T.K.Bera, B.G.Chandresh

Abstract:

The scroll pump belongs to the category of positive displacement pump can be used for continuous pumping of gases at low pressure apart from general vacuum application. The shape of volume occupied by the gas moves and deforms continuously as the spiral orbits. To capture flow features in such domain where mesh deformation varies with time in a complicated manner, mesh less solver was found to be very useful. Least Squares Kinetic Upwind Method (LSKUM) is a kinetic theory based mesh free Euler solver working on arbitrary distribution of points. Here upwind is enforced in molecular level based on kinetic flux vector splitting scheme (KFVS). In the present study we extended the LSKUM to moving node viscous flow application. This new code LSKUM-NS-MN for moving node viscous flow is validated for standard airfoil pitching test case. Simulation performed for flow through scroll pump using LSKUM-NS-MN code agrees well with the experimental pumping speed data.

Keywords: Least Squares, Moving node, Pitching, Spirals.

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Authors: Kyoungwoo Park, Ji-Won Han, Hyo-Jae Lim, Byeong-Sam Kim, Juhee Lee

Abstract:

Shape optimization of the airfoil with high aspect ratio of long endurance unmanned aerial vehicle (UAV) is performed by the multi-objective optimization technology coupled with computational fluid dynamics (CFD). For predicting the aerodynamic characteristics around the airfoil the high-fidelity Navier-Stokes solver is employed and SMOGA (Simple Multi-Objective Genetic Algorithm), which is developed by authors, is used for solving the multi-objective optimization problem. To obtain the optimal solutions of the design variable (i.e., sectional airfoil profile, wing taper ratio and sweep) for high performance of UAVs, both the lift and lift-to-drag ratio are maximized whereas the pitching moment should be minimized, simultaneously. It is found that the lift force and lift-to-drag ratio are linearly dependent and a unique and dominant solution are existed. However, a trade-off phenomenon is observed between the lift-to-drag ratio and pitching moment. As the result of optimization, sixty-five (65) non-dominated Pareto individuals at the cutting edge of design spaces that is decided by airfoil shapes can be obtained.

Keywords: Unmanned aerial vehicle (UAV), Airfoil, CFD, Shape optimization, Lift-to-drag ratio.

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Authors: Yiming Jin, Yuanhao Gao

Abstract:

In this paper, using the method of multiple scales, the second sub-harmonic resonance in vortex-induced vibrations (VIV) of a marine pipeline close to the seabed is investigated based on a developed wake oscillator model. The amplitude-frequency equations are also derived. It is found that the oscillation will increase all the time when both discriminants of the amplitude-frequency equations are positive while the oscillation will decay when the discriminants are negative.

Keywords: Vortex-induced vibrations, marine pipeline, seabed, sub-harmonic resonance.

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Authors: Koson Pitaksuttayaprot, Winai Jaikla

Abstract:

The realization of current-mode quadrature oscillators using current controlled current conveyor transconductance amplifiers (CCCCTAs) and grounded capacitors is presented. The proposed oscillators can provide 2 sinusoidal output currents with 90º phase difference. It is enabled non-interactive dual-current control for both the condition of oscillation and the frequency of oscillation. High output impedances of the configurations enable the circuit to be cascaded without additional current buffers. The use of only grounded capacitors is ideal for integration. The circuit performances are depicted through PSpice simulations, they show good agreement to theoretical anticipation.

Keywords: Current-mode, Oscillator, Integrated circuit, CCCCTA.

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Authors: Eduardo Romero, Marcelo Costamagna, Gabriela Peretti, Carlos Marqués

Abstract:

This work evaluates the ability of OBT for detecting parametric faults in continuous-time filters. To this end, we adopt two filters with quite different topologies as cases of study and a previously reported statistical fault model. In addition, we explore the behavior of the test schemes when a particular test condition is changed. The new data reported here, obtained from a fault simulation process, reveal a lower performance of OBT not observed in previous work using single-deviation faults, even under the change in the test condition.

Keywords: Testing, analog fault simulation, analog filter test, oscillation based test.

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Authors: Prungsak Uttaphut

Abstract:

An implementation of current-mode multiphase sinusoidal oscillators is presented. Using CFTA-based lossy integrators, odd and odd/even phase systems can be realized with following advantages. The condition of oscillation and frequency of oscillation can be orthogonally tuned. The high output impedances facilitate easy driving an external load without additional current buffers. The proposed MSOs provide odd or even phase signals that are equally spaced in phase and equal amplitude. The circuit requires one CFTA, one resistor and one grounded capacitor per phase without additional current amplifier. The results of PSPICE simulations using CMOS CFTA are included to verify theory.

Keywords: multiphase sinusoidal oscillator, current-mode, CFTA, lossy integrator

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Authors: A. Hamdoune, M. Abdelmoumene, A. Hamroun

Abstract:

The objective of this paper is to simulate static I-V and dynamic characteristics of an appropriated and recessed n-GaN/AlxGa1-xN/GaN high electron mobility (HEMT). Using SILVACO TCAD device simulation, and optimized technological parameters; we calculate the drain-source current (lDS) as a function of the drain-source voltage (VDS) for different values ​​of the gate-source voltage (VGS), and the drain-source current (lDS) depending on the gate-source voltage (VGS) for a drain-source voltage (VDS) of 20 V, for various temperatures. Then, we calculate the cut-off frequency and the maximum oscillation frequency for different temperatures.

We obtain a high drain-current equal to 60 mA, a low knee voltage (Vknee) of 2 V, a high pinch-off voltage (VGS0) of 53.5 V, a transconductance greater than 600 mS/mm, a cut-off frequency (f_T) of about 330 GHz, and a maximum oscillation frequency (fmax) of about 1 THz.

Keywords: n-GaN/AlGaN/GaN HEMT, drain-source current (IDS), transconductance (gm), cut-off frequency (fT), maximum oscillation frequency (fmax).

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Authors: Ermanu A. Hakim, Adi Soeprijanto, Mauridhi H.P

Abstract:

This paper presents PSS (Power system stabilizer) design based on optimal fuzzy PID (OFPID). OFPID based PSS design is considered for single-machine power systems. The main motivation for this design is to stabilize or to control low-frequency oscillation on power systems. Firstly, describing the linear PID control then to combine this PID control with fuzzy logic control mechanism. Finally, Fuzzy PID parameters (Kp. Kd, KI, Kupd, Kui) are tuned by Genetic Algorthm (GA) to reach optimal global stability. The effectiveness of the proposed PSS in increasing the damping of system electromechanical oscillation is demonstrated in a one-machine-infinite-bus system

Keywords: Fuzzy PID, Genetic Algorithm, power system stabilizer.

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Authors: Mohammad Taghi Darvishi, Samad Kheybari

Abstract:

In this article, we are dealing with a model consisting of a classical Van der Pol oscillator coupled gyroscopically to a linear oscillator. The major problem is analyzed. The regular dynamics of the system is considered using analytical methods. In this case, we provide an approximate solution for this system using parameter-expansion method. Also, we find approximate values for frequencies of the system. In parameter-expansion method the solution and unknown frequency of oscillation are expanded in a series by a bookkeeping parameter. By imposing the non-secularity condition at each order in the expansion the method provides different approximations to both the solution and the frequency of oscillation. One iteration step provides an approximate solution which is valid for the whole solution domain.

Keywords: Parameter-expansion method, classical Van der Pol oscillator.

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