Lab 4: Prelab
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The Damped Harmonic Oscillator
THE DAMPED HARMONIC OSCILLATOR Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8 Free, undamped oscillators – other examples k m L No friction I C k m q 1 x m!x! = !kx q!! = ! q LC ! ! r; r L = θ Common notation for all g !! 2 T ! " # ! !!! + " ! = 0 m L 0 mg k friction m 1 LI! + q + RI = 0 x C 1 Lq!!+ q + Rq! = 0 C m!x! = !kx ! bx! ! r L = cm θ Common notation for all g !! ! 2 T ! " # ! # b'! !!! + 2"!! +# ! = 0 m L 0 mg Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ ! !kx ! k Need a model for this. m Try restoring force proportional to velocity k m x !bx! How do we choose a model? Physically reasonable, mathematically tractable … Validation comes IF it describes the experimental system accurately Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ !kx ! bx! = m!x! ! Divide by coefficient of d2x/dt2 ! and rearrange: x 2 x 2 x 0 !!+ ! ! + " 0 = inverse time β and ω0 (rate or frequency) are generic to any oscillating system This is the notation of TM; Main uses γ = 2β. Natural motion of damped harmonic oscillator 2 x˙˙ + 2"x˙ +#0 x = 0 Try x(t) = Ce pt C, p are unknown constants ! x˙ (t) = px(t), x˙˙ (t) = p2 x(t) p2 2 p 2 x(t) 0 Substitute: ( + ! + " 0 ) = ! 2 2 Now p is known (and p = !" ± " ! # 0 there are 2 p values) p t p t x(t) = Ce + + C'e " Must be sure to make x real! ! Natural motion of damped HO Can identify 3 cases " < #0 underdamped ! " > #0 overdamped ! " = #0 critically damped time ---> ! underdamped " < #0 # 2 !1 = ! 0 1" 2 ! 0 ! time ---> 2 2 p = !" ± " ! # 0 = !" ± i#1 x(t) = Ce"#t+i$1t +C*e"#t"i$1t Keep x(t) real "#t x(t) = Ae [cos($1t +%)] complex <-> amp/phase System oscillates at "frequency" ω1 (very close to ω0) ! - but in fact there is not only one single frequency associated with the motion as we will see. -
The Q of Oscillators
The Q of oscillators References: L.R. Fortney – Principles of Electronics: Analog and Digital, Harcourt Brace Jovanovich 1987, Chapter 2 (AC Circuits) H. J. Pain – The Physics of Vibrations and Waves, 5th edition, Wiley 1999, Chapter 3 (The forced oscillator). Prerequisite: Currents through inductances, capacitances and resistances – 2nd year lab experiment, Department of Physics, University of Toronto, http://www.physics.utoronto.ca/~phy225h/currents-l-r-c/currents-l-c-r.pdf Introduction In the Prerequisite experiment, you studied LCR circuits with different applied signals. A loop with capacitance and inductance exhibits an oscillatory response to a disturbance, due to the oscillating energy exchange between the electric and magnetic fields of the circuit elements. If a resistor is added, the oscillation will become damped. 1 In this experiment, the LCR circuit will be driven at resonance frequencyωr = , when the LC transfer of energy between the driving source and the circuit will be a maximum. LCR circuits at resonance. The transfer function. Ohm’s Law applied to a LCR loop (Figure 1) can be written in complex notation (see Appendix from the Prerequisite) Figure 1 LCR circuit for resonance studies v( jω) Ohm’s Law: i( jω) = (1) Z - 1 - i(jω) and v(jω) are complex instantaneous values of current and voltage, ω is the angular frequency (ω = 2πf ) , Z is the complex impedance of the loop: 1 Z = R + jωL − (2) ωC j = −1 is the complex number The voltage across the resistor from Figure 1, as a result of current i can be expressed as: vR ( jω) = Ri( jω) (3) Eliminating i(jω) from Equations (1) and (3) results into: R v ( jω) = v( jω) (4) R R + j(ωL −1/ωC) Equation (4) can be put into the general form: vR ( jω) = H ( jω)v( jω) (5) where H(jω) is called a transfer function across the resistor, in the frequency domain. -
Frequency Response
EE105 – Fall 2015 Microelectronic Devices and Circuits Frequency Response Prof. Ming C. Wu [email protected] 511 Sutardja Dai Hall (SDH) Amplifier Frequency Response: Lower and Upper Cutoff Frequency • Midband gain Amid and upper and lower cutoff frequencies ωH and ω L that define bandwidth of an amplifier are often of more interest than the complete transferfunction • Coupling and bypass capacitors(~ F) determineω L • Transistor (and stray) capacitances(~ pF) determineω H Lower Cutoff Frequency (ωL) Approximation: Short-Circuit Time Constant (SCTC) Method 1. Identify all coupling and bypass capacitors 2. Pick one capacitor ( ) at a time, replace all others with short circuits 3. Replace independent voltage source withshort , and independent current source withopen 4. Calculate the resistance ( ) in parallel with 5. Calculate the time constant, 6. Repeat this for each of n the capacitor 7. The low cut-off frequency can be approximated by n 1 ωL ≅ ∑ i=1 RiSCi Note: this is an approximation. The real low cut-off is slightly lower Lower Cutoff Frequency (ωL) Using SCTC Method for CS Amplifier SCTC Method: 1 n 1 fL ≅ ∑ 2π i=1 RiSCi For the Common-Source Amplifier: 1 # 1 1 1 & fL ≅ % + + ( 2π $ R1SC1 R2SC2 R3SC3 ' Lower Cutoff Frequency (ωL) Using SCTC Method for CS Amplifier Using the SCTC method: For C2 : = + = + 1 " 1 1 1 % R3S R3 (RD RiD ) R3 (RD ro ) fL ≅ $ + + ' 2π # R1SC1 R2SC2 R3SC3 & For C1: R1S = RI +(RG RiG ) = RI + RG For C3 : 1 R2S = RS RiS = RS gm Design: How Do We Choose the Coupling and Bypass Capacitor Values? • Since the impedance of a capacitor increases with decreasing frequency, coupling/bypass capacitors reduce amplifier gain at low frequencies. -
Arxiv:1704.05328V1 [Cond-Mat.Mes-Hall] 18 Apr 2017 Tuning the Mechanical Mode Frequencies by More Than Silicon Wafers
Quantitative Determination of the Mechanical Properties of Nanomembrane Resonators by Vibrometry In Continuous Light Fan Yang,∗ Reimar Waitz,y and Elke Scheer Department of Physics, Universit¨atKonstanz, 78464 Konstanz, Germany We present an experimental study of the bending waves of freestanding Si3N4 nanomembranes using optical profilometry in varying environments such as pressure and temperature. We introduce a method, named Vibrometry in Continuous Light (VICL) that enables us to disentangle the response of the membrane from the one of the excitation system, thereby giving access to the eigenfrequency and the quality (Q) factor of the membrane by fitting a model of a damped driven harmonic oscillator to the experimental data. The validity of particular assumptions or aspects of the model such as damping mechanisms, can be tested by imposing additional constraints on the fitting procedure. We verify the performance of the method by studying two modes of a 478 nm thick Si3N4 freestanding membrane and find Q factors of 2 × 104 for both modes at room temperature. Finally, we observe a linear increase of the resonance frequency of the ground mode with temperature which amounts to 550 Hz=◦C for a ground mode frequency of 0:447 MHz. This makes the nanomembrane resonators suitable as high-sensitive temperature sensors. I. INTRODUCTION frequencies of bending waves of nanomembranes may range from a few kHz to several 100 MHz [6], those of Nanomechanical membranes are extensively used in a thickness oscillation may even exceed 100 GHz [21], re- variety of applications including among others high fre- quiring a versatile excitation and detection method able quency microwave devices [1], human motion detectors to operate in this wide frequency range and to resolve [2], and gas sensors [3]. -
Feedback Amplifiers
UNIT II FEEDBACK AMPLIFIERS & OSCILLATORS FEEDBACK AMPLIFIERS: Feedback concept, types of feedback, Amplifier models: Voltage amplifier, current amplifier, trans-conductance amplifier and trans-resistance amplifier, feedback amplifier topologies, characteristics of negative feedback amplifiers, Analysis of feedback amplifiers, Performance comparison of feedback amplifiers. OSCILLATORS: Principle of operation, Barkhausen Criterion, types of oscillators, Analysis of RC-phase shift and Wien bridge oscillators using BJT, Generalized analysis of LC Oscillators, Hartley and Colpitts’s oscillators with BJT, Crystal oscillators, Frequency and amplitude stability of oscillators. 1.1 Introduction: Feedback Concept: Feedback: A portion of the output signal is taken from the output of the amplifier and is combined with the input signal is called feedback. Need for Feedback: • Distortion should be avoided as far as possible. • Gain must be independent of external factors. Concept of Feedback: Block diagram of feedback amplifier consist of a basic amplifier, a mixer (or) comparator, a sampler, and a feedback network. Figure 1.1 Block diagram of an amplifier with feedback A – Gain of amplifier without feedback. A = X0 / Xi Af – Gain of amplifier with feedback.Af = X0 / Xs β – Feedback ratio. β = Xf / X0 X is either voltage or current. 1.2 Types of Feedback: 1. Positive feedback 2. Negative feedback 1.2.1 Positive Feedback: If the feedback signal is in phase with the input signal, then the net effect of feedback will increase the input signal given to the amplifier. This type of feedback is said to be positive or regenerative feedback. Xi=Xs+Xf Af = = = Af= Here Loop Gain: The product of open loop gain and the feedback factor is called loop gain. -
Unit I Microwave Transmission Lines
UNIT I MICROWAVE TRANSMISSION LINES INTRODUCTION Microwaves are electromagnetic waves with wavelengths ranging from 1 mm to 1 m, or frequencies between 300 MHz and 300 GHz. Apparatus and techniques may be described qualitatively as "microwave" when the wavelengths of signals are roughly the same as the dimensions of the equipment, so that lumped-element circuit theory is inaccurate. As a consequence, practical microwave technique tends to move away from the discrete resistors, capacitors, and inductors used with lower frequency radio waves. Instead, distributed circuit elements and transmission-line theory are more useful methods for design, analysis. Open-wire and coaxial transmission lines give way to waveguides, and lumped-element tuned circuits are replaced by cavity resonators or resonant lines. Effects of reflection, polarization, scattering, diffraction, and atmospheric absorption usually associated with visible light are of practical significance in the study of microwave propagation. The same equations of electromagnetic theory apply at all frequencies. While the name may suggest a micrometer wavelength, it is better understood as indicating wavelengths very much smaller than those used in radio broadcasting. The boundaries between far infrared light, terahertz radiation, microwaves, and ultra-high-frequency radio waves are fairly arbitrary and are used variously between different fields of study. The term microwave generally refers to "alternating current signals with frequencies between 300 MHz (3×108 Hz) and 300 GHz (3×1011 Hz)."[1] Both IEC standard 60050 and IEEE standard 100 define "microwave" frequencies starting at 1 GHz (30 cm wavelength). Electromagnetic waves longer (lower frequency) than microwaves are called "radio waves". Electromagnetic radiation with shorter wavelengths may be called "millimeter waves", terahertz radiation or even T-rays. -
Wave Guides & Resonators
UNIT I WAVEGUIDES & RESONATORS INTRODUCTION Microwaves are electromagnetic waves with wavelengths ranging from 1 mm to 1 m, or frequencies between 300 MHz and 300 GHz. Apparatus and techniques may be described qualitatively as "microwave" when the wavelengths of signals are roughly the same as the dimensions of the equipment, so that lumped-element circuit theory is inaccurate. As a consequence, practical microwave technique tends to move away from the discrete resistors, capacitors, and inductors used with lower frequency radio waves. Instead, distributed circuit elements and transmission-line theory are more useful methods for design, analysis. Open-wire and coaxial transmission lines give way to waveguides, and lumped-element tuned circuits are replaced by cavity resonators or resonant lines. Effects of reflection, polarization, scattering, diffraction, and atmospheric absorption usually associated with visible light are of practical significance in the study of microwave propagation. The same equations of electromagnetic theory apply at all frequencies. While the name may suggest a micrometer wavelength, it is better understood as indicating wavelengths very much smaller than those used in radio broadcasting. The boundaries between far infrared light, terahertz radiation, microwaves, and ultra-high-frequency radio waves are fairly arbitrary and are used variously between different fields of study. The term microwave generally refers to "alternating current signals with frequencies between 300 MHz (3×108 Hz) and 300 GHz (3×1011 Hz)."[1] Both IEC standard 60050 and IEEE standard 100 define "microwave" frequencies starting at 1 GHz (30 cm wavelength). Electromagnetic waves longer (lower frequency) than microwaves are called "radio waves". Electromagnetic radiation with shorter wavelengths may be called "millimeter waves", terahertz Page 1 radiation or even T-rays. -
RF-MEMS Load Sensors with Enhanced Q-Factor and Sensitivity In
Microelectronic Engineering 88 (2011) 247–253 Contents lists available at ScienceDirect Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee RF-MEMS load sensors with enhanced Q-factor and sensitivity in a suspended architecture ⇑ Rohat Melik a, Emre Unal a, Nihan Kosku Perkgoz a, Christian Puttlitz b, Hilmi Volkan Demir a, a Departments of Electrical Engineering and Physics, Nanotechnology Research Center, and Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey b Department of Mechanical Engineering, Orthopaedic Bioengineering Research Laboratory, Colorado State University, Fort Collins, CO 80523, USA article info abstract Article history: In this paper, we present and demonstrate RF-MEMS load sensors designed and fabricated in a suspended Received 31 August 2009 architecture that increases their quality-factor (Q-factor), accompanied with an increased resonance fre- Received in revised form 7 July 2010 quency shift under load. The suspended architecture is obtained by removing silicon under the sensor. Accepted 29 October 2010 We compare two sensors that consist of 195 lm  195 lm resonators, where all of the resonator features Available online 9 November 2010 are of equal dimensions, but one’s substrate is partially removed (suspended architecture) and the other’s is not (planar architecture). The single suspended device has a resonance of 15.18 GHz with 102.06 Q-fac- Keywords: tor whereas the single planar device has the resonance at 15.01 GHz and an associated Q-factor of 93.81. Fabrication For the single planar device, we measured a resonance frequency shift of 430 MHz with 3920 N of applied IC Resonance frequency shift load, while we achieved a 780 MHz frequency shift in the single suspended device. -
Waveguides Waveguides, Like Transmission Lines, Are Structures Used to Guide Electromagnetic Waves from Point to Point. However
Waveguides Waveguides, like transmission lines, are structures used to guide electromagnetic waves from point to point. However, the fundamental characteristics of waveguide and transmission line waves (modes) are quite different. The differences in these modes result from the basic differences in geometry for a transmission line and a waveguide. Waveguides can be generally classified as either metal waveguides or dielectric waveguides. Metal waveguides normally take the form of an enclosed conducting metal pipe. The waves propagating inside the metal waveguide may be characterized by reflections from the conducting walls. The dielectric waveguide consists of dielectrics only and employs reflections from dielectric interfaces to propagate the electromagnetic wave along the waveguide. Metal Waveguides Dielectric Waveguides Comparison of Waveguide and Transmission Line Characteristics Transmission line Waveguide • Two or more conductors CMetal waveguides are typically separated by some insulating one enclosed conductor filled medium (two-wire, coaxial, with an insulating medium microstrip, etc.). (rectangular, circular) while a dielectric waveguide consists of multiple dielectrics. • Normal operating mode is the COperating modes are TE or TM TEM or quasi-TEM mode (can modes (cannot support a TEM support TE and TM modes but mode). these modes are typically undesirable). • No cutoff frequency for the TEM CMust operate the waveguide at a mode. Transmission lines can frequency above the respective transmit signals from DC up to TE or TM mode cutoff frequency high frequency. for that mode to propagate. • Significant signal attenuation at CLower signal attenuation at high high frequencies due to frequencies than transmission conductor and dielectric losses. lines. • Small cross-section transmission CMetal waveguides can transmit lines (like coaxial cables) can high power levels. -
Teaching the Difference Between Stiffness and Damping
LECTURE NOTES « Teaching the Difference Between Stiffness and Damping Henry C. FU and Kam K. Leang In contrast to the parallel spring-mass-damper, in the necessary step in the design of an effective control system series mass-spring-damper system the stiffer the system, A is to understand its dynamics. It is not uncommon for a the more dissipative its behavior, and the softer the sys- well-trained control engineer to use such an understanding to tem, the more elastic its behavior. Thus teaching systems redesign the system to become easier to control, which requires modeled by series mass-spring-damper systems allows an even deeper understanding of how the components of a students to appreciate the difference between stiffness and system influence its overall dynamics. In this issue Henry C. damping. Fu and Kam K. Leang discuss the diff erence between stiffness To get students to consider the difference between soft and damping when understanding the dynamics of mechanical and damped, ask them to consider the following scenario: systems. suppose a bullfrog is jumping to land on a very slippery floating lily pad as illustrated in Figure 2 and does not want to fall off. The frog intends to hit the flower (but he simple spring, mass, and damper system is ubiq- cannot hold onto it) in the center to come to a stop. If the uitous in dynamic systems and controls courses [1]. TThis column considers a concept students often have trouble with: the difference between a “soft” system, which has a small elastic restoring force, and a “damped” system, x c which dissipates energy quickly. -
DETERMINATION of the APPROPRIATE CUTOFF FREQUENCY in the DIGITAL FILTER DATA SMOOTHING PROCEDURE By
'DETERMINATION OF THE APPROPRIATE CUTOFF FREQUENCY IN THE DIGITAL FILTER DATA SMOOTHING PROCEDURE by BING YU B.S., Peking Institute of Physical Education, 1982 A MASTER'S THESIS Submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Physical Education and Leisure Studies KANSAS STATE UNIVERSITY 1988 Approved by: Major Professo 3# AllSDfl 5327b7 ']cl ACKNOWLEDGEMENTS The author wishes to acknowledge the assistance and support of the entire graduate faculty of Kansas State University's Department of Physical Education and Leisure Studies. Special thanks go to committee members Dr. Stephan Konz and Dr. Kathleen Williams for their unique perspectives and editorial assistance. Most of all, I would like to thank my major professor, Dr. Larry Noble, for his integrity, his enthusiasm for knowledge, and the tremendous amount of time and assistance he has given me over the past two years. 11 DEDICATION This thesis is dedicated to my parents, Dr. Gou-Rei Yu and Ming-Hua Lu, to my wife Wei Li, to her parents, Dr. Ping Li and Dr. Xiu-Zhang Yu, and to all of the other folks of my family and her family for their understanding of my absence when my son Charlse Alan Yu was born. Their constant support and encouragement are deeply appreciated. 111 . CONTENTS ACKNOWLEDGMENTS ii DEDICATION iii LIST OF FIGURES vi LIST OF TABLES ix Chapter 1 INTRODUCTION 1 Statement Of The Problem 2 Definitions 3 2 REVIEW OF RELATED LITERATURE 7 The Nature Of Errors 7 Sources Of Errors 10 Data Smoothing Techniques Used In Sport Biomechanics 15 Finite difference technique 15 Least square polynomial approxination. -
Modified Chebyshev-2 Filters with Low Q-Factors
J KAU: Eng. Sci., vol. 3, pp. 21-34 (1411 A.H.l1991 A.D.) Modified Chebyshev-2 Filters with Low Q-Factors A. M. MILYANI and A. M. AFFANDI Department ofElectrical and Computer Engineering, Faculty ofEngineering, King Abdulaziz University, Jeddah, Saudi Arabia. ABSTRACT A modified low pass maximally flat inverse Chebyshev filter is suggested in this paper, and is shown to have improvement over previous known Chebyshev filters. The coefficients of this modified filter, using a higher order polynomial with multiplicity of the dominant pole-pair, have been determined. The pole locations and the Q factors for different orders of filters are tabulated using an optimization algorithm. 1. Introduction Recently, there has been a great deal of interest in deriving suboptimal transfer func tions with low Q dominant poles[1-3J. The suboptimality of these functions allows for low precision requirements in both active RC-filters and digital filters. It is, however, noted that the minimal order filter is not necessarily the least complex filter. In a recent article, Premoli[11 used the notion of multiplicity in the dominant poles to obtain multiple critical root maximally flat (MUCROMAF) polynomials for low pass filters. An alternative method for deriving these functions, referred to as mod ified Butterworth functions, has been presented by Massad and Yariagaddal2J . Also a new class of multiple critical root pair, equal ripple (MUCROER) filtering func tions, having higher degree than the filtering functions, has been presented by Premoli[1l. By relaxing the equal ripple conditions, Massad and Yarlagadda derived a new algorithm to find modified Chebyshev functions which have lower Qdominant polesl21 .