DOCSLIB.ORG

Sinusoidal Signals Sinusoidal Signals

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sinusoidal Signals Sinusoidal Signals SINUSOIDAL SIGNALS SINUSOIDAL SIGNALS x(t) = cos(2¼ft + ') = cos(!t + ') (continuous time) x[n] = cos(2¼fn + ') = cos(!n + ') (discrete time) f : frequency (s¡1 (Hz)) ! : angular frequency (radians/s) ' : phase (radians) 2 Sinusoidal signals A cos(ω t) A sin(ω t) A 0 −A 0 T/2 T 3T/2 2T xc(t) = A cos(2¼ft) xs(t) = A sin(2¼ft) = A cos(2¼ft ¡ ¼=2) - amplitude A -amplitude A - period T = 1=f -period T = 1=f - phase: 0 -phase: ¡¼=2 3 WHY SINUSOIDAL SIGNALS? ² Physical reasons: - harmonic oscillators generate sinusoids, e.g., vibrating structures - waves consist of sinusoidals, e.g., acoustic waves or electromagnetic waves used in wireless transmission ² Psychophysical reason: - speech consists of superposition of sinusoids - human ear detects frequencies - human eye senses light of various frequencies ² Mathematical (and physical) reason: - Linear systems, both physical systems and man-made ¯lters, a®ect a signal frequency by frequency (hence low- pass, high-pass etc ¯lters) 4 EXAMPLE: TRANSMISSION OF A LOW-FREQUENCY SIGNAL USING HIGH-FREQUENCY ELECTROMAGNETIC (RADIO) SIGNAL - A POSSIBLE (CONVENTIONAL) METHOD: AMPLITUDE MODULATION (AM) Example: low-frequency signal v(t) = 5 + 2 cos(2¼f¢t); f¢ = 20 Hz High-frequency carrier wave vc(t) = cos(2¼fct); fc = 200 Hz Amplitude modulation (AM) of carrier (electromagnetic) wave: x(t) = v(t) cos(2¼fct) 5 8 6 4 2 v 0 −2 −4 −6 −8 8 6 4 2 x 0 −2 −4 −6 −8 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 t Top: v(t) (dashed) and vc(t) = cos(2¼fct). Bottom: transmitted signal x(t) = v(t) cos(2¼fct). 6 Frequency contents of transmitted signal x(t) = v(t) cos(2¼fct) = (5 + 2 cos(2¼f¢t)) cos(2¼fct) = 5 cos(2¼fct) + 2 cos(2¼f¢t) cos(2¼fct) Trigonometric identity: 1 1 cos ® cos ¯ = cos(® ¡ ¯) + cos(® + ¯) 2 2 ) 2 cos(2¼f¢t) cos(2¼fct) = cos (2¼(fc ¡ f¢)t)+cos (2¼(fc + f¢)t) 7 ) x(t) = 5 cos(2¼fct) + cos (2¼(fc ¡ f¢)t) + cos (2¼(fc + f¢)t) 5 Spectrum Spectrum of v(t) of vc(t) 2 1 - 0 20 200 Frequency Spectrum of x(t) 5 1 1 - 0 180 200 220 Frequency 8 We see that a low-frequency signal in frequency range 0 · fs · fmax (baseband signal) can be transmitted as a signal in the frequency range fc ¡ fmax · f · fc ¡ fmax ("RF" (radio frequency) signal). Another analog modulation technique is frequency modulation (FM) 9 Representation of sinusoidal signal A. Amplitude and phase B. Sine and cosine components C. Complex exponentials 10 A. Amplitude and phase x(t) = A cos(2¼ft + ') or x(t) = A cos(!t + ') where ! = 2¼f is the angular frequency (radians/second) The phase ' describes translation in time compared to cos(!t): x(t) = A cos(!t + ') = A cos(!(t + '=!)) 11 B. Sine and cosine components x(t) = a cos(!t) + b sin(!t) Follows from trigonometric identity: cos(® + ¯) = cos(®) cos(¯) ¡ sin(®) sin(¯) ) x(t) = A cos(!t + ') = A cos(') cos(!t) ¡ A sin(') sin(!t) Hence: a = A cos('); b = ¡A sin(') and, conversely, p A = a2 + b2;' = ¡ arctan(b=a) 12 EXAMPLE x(t) 1 0 −1 −2 0 2 4 6 8 10 12 Signal x(t) = cos(!t + ') Period: T = 10 (s) Frequency: f = 1=T = 0:1 (s¡1 =Hz (periods/second)) Angular frequency: ! = 2¼f = 0:628 (radians/s) 13 A. Representation using amplitude and phase Amplitude: A = 1 From ¯gure we see that x(t) = cos(!(t ¡ 2)). But x(t) = cos(!t + ') = cos(!(t + '=!) ) Phase: ' = ¡2! = ¡0:4¼ radians or ¡0:4 ¢ 180=¼ = ¡72± 14 B. Sine and cosine components x(t) = a cos(!t) + b sin(!t) where a = cos(') = cos(¡0:4¼) = 0:3090 b = ¡ sin(') = ¡ sin(¡0:4¼) = 0:9511 1 x(t) 0 −1 −2 0 2 4 6 8 10 12 1 Cosine component Sine component 0 −1 −2 0 2 4 6 8 10 12 x(t) = 0:3090 cos(!t) + 0:9511 sin(!t) 15 C. Representation using complex exponentials Background. A common operation is to combine sinusoidal components with di®erent phases: x(t) = A1 cos(!t + '1) + A2 cos(!t + '2) One way: ¯rst write A1 cos(!t + '1) = a1 cos(!t) + b1 sin(!t) A2 cos(!t + '2) = a2 cos(!t) + b2 sin(!t) Then x(t) = (a1 + a2) cos(!t) + (b1 + b2) sin(!t) 16 Streamlining the above procedure We have: cos(!t + ') = cos(') cos(!t) ¡ sin(') sin(!t) and sin(!t + ') = sin(') cos(!t) + cos(') sin(!t) Instead of working with the two signals cos(!t) and sin(!t) it is convenient to working with the single complex-valued signal g(!t) = cos(!t) + j sin(!t) p where j = ¡1 (imaginary unit number) 17 Im 6 g(!t) = cos(!t) + j sin(!t) j sin(!t) ©©* ©© ©© ©© 1 ©© ©© ©© ©© ©© ©© !t ©© - cos(!t) Re Then: g(!t + ') = cos(!t + ') + j sin(!t + ') = cos(') cos(!t) ¡ sin(') sin(!t) ¡ ¢ +j sin(') cos(!t) + cos(') sin(!t) ¡ ¢¡ ¢ = cos(') + j sin(') cos(!t) + j sin(!t) = g(')g(!t) 18 In terms of the complex-valued signal, phase shift with angle ' is equivalent to multiplication by the complex number g('): g(!t + ') = g(')g(!t)(A) ) cos(!t + ') = Re (g(')g(!t)) sin(!t + ') = Im (g(')g(!t)) We will see that using the multiplication (A) to describe phase shifts results in signi¯cant simpli¯cation of formulae and calculations 19 Properties of the complex function g(θ) = cos(θ) + j sin(θ): - g(0) = 1 - g(θ1 + θ2) = g(θ1)g(θ2) dg(θ) - dθ = ¡ sin(θ) + j cos(θ) = jg(θ) The only function which satis¯es these relations is g(θ) = ejθ e = 2:71828182845904523536 ::: (Euler's number) 20 Hence we have: Euler's relation ejθ = cos(θ) + j sin(θ) e = 2:71828182845904523536 ::: (Euler's number) As e¡jθ = cos(θ) ¡ j sin(θ) we have 1 ¡ ¢ cos(θ) = ejθ + e¡jθ 2 1 ¡ ¢ sin(θ) = ejθ ¡ e¡jθ 2j 21 It follows that x(t) = a cos(!t) + b sin(!t) µ ¶ µ ¶ a b a b = + ej!t + ¡ e¡j!t 2 2j 2 2j 1 1 = (a ¡ jb) ej!t + (a + jb) e¡j!t 2 2 or x(t) = c ej!t + c¤e¡j!t 1 ¤ 1 where c = 2 (a ¡ jb) and c = 2 (a + jb) (complex conjugate) Thus the cosine and sine components are parameterized as the real and imaginary components of the complex number c 22 PREVIOUS EXAMPLE: The signal x(t) = cos(!t ¡ 0:4¼) = a cos(!t) + b sin(!t) = 0:3090 cos(!t) + 0:9511 sin(!t) can be represented using complex exponentials as x(t) = c ej!t + c¤ e¡j!t where 1 c = (a ¡ jb) = 0:1545 ¡ j0:4755 2 1 c¤ = (a + jb) = 0:1545 + j0:4755 2 Hence: x(t) = (0:1545 ¡ j0:4755) ej!t + (0:1545 + j0:4755) e¡j!t 23 SUMMARY Let us summarize the results so far. We have the following alternative representations of a sinusoidal signal: ² Amplitude and phase: x(t) = A cos(!t + ') ² Sine and cosine components: x(t) = a cos(!t) + b sin(!t) where a = A cos('); b = ¡A sin(') or p A = a2 + b2;' = ¡ arctan(b=a) 24 ² Complex exponentials: x(t) = c ej!t + c¤ e¡j!t where 1 1 c = (a ¡ jb) ; c¤ = (a + jb) 2 2 or a = 2Re(c); b = ¡2Im(c) 25 EXAMPLE - Signal transmission with reflected component Transmitted signal: x(t) = cos(!t) Received signal: y(t) = x(t) + rx(t ¡ ¿) PROBLEM: Determine amplitude and phase of y(t) We have y(t) = cos(!t) + r cos(!(t ¡ ¿)) The problem can be solved in a straightforward way by introducing complex exponential: 1 ¡ ¢ x(t) = ej!t + e¡j!t 2 26 Then: y(t) = cos(!t) + r cos(!(t ¡ ¿)) 1 ¡ ¢ 1 ³ ´ = ej!t + e¡j!t + r ej!(t¡¿) + e¡j!(t¡¿) 2 2 1 ¡ ¢ 1 ¡ ¢ = 1 + re¡j!¿ ej!t + 1 + rej!¿ e¡j!t 2 2 Here, 1 ¡ ¢ 1³ ´ 1 + re¡j!¿ = 1 + r cos(!¿) ¡ jr sin(!¿) 2 2 and 1 ¡ ¢ 1³ ´ 1 + rej!¿ = 1 + r cos(!¿) + jr sin(!¿) 2 2 27 Then, de¯ning a = 1 + r cos(!¿); b = r sin(!¿) we have (cf. above) 1 1 y(t) = (a ¡ jb) ej!t + (a + jb) e¡j!t 2 2 = a cos(!t) + b sin(!t) = Ay cos(!t + 'y) where p 2 2 Ay = a + b ;'y = ¡ arctan(b=a) 28 SOME SIMPLIFICATIONS As 1 1 a cos(!t) + b sin(!t) = (a ¡ jb) ej!t + (a + jb) e¡j!t 2 2 it follows that it is su±cient to consider the complex-valued signal 1 (a ¡ jb) ej!t 2 Here, the complex number 1 c = (a ¡ jb) 2 determines the signal's amplitude and phase. 29 To see how this works, consider a linear dynamical system with input x(t) and output y(t) x(t) - G - y(t) For the sinusoidal input x(t) = cos(!t) the output takes the form y(t) = A cos(!t + ') = a cos(!t) + b sin(!t) and we want to determine the amplitude A and phase '. 30 As the phase shift introduces a combination of cosines and sines, the problem can be simpli¯ed by embedding the input signal into a larger class of signals involving both a cosine and a sine component. It turns out that a convenient way to do this is to consider the complex input signal xc(t) = cos(!t) + j sin(!t) = ej!t The output then also has real and imaginary components: yc(t) = A cos(!t + ') + jA sin(!t + ') = Aej(!t+') = Aej'ej!t 31 or j!t yc(t) = c e = c xc(t) where c = Aej'.
Load more

Recommended publications
  • Glossary Physics (I-Introduction)
    1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
    [Show full text]
  • Frequency Response = K − Ml
    Frequency Response 1. Introduction We will examine the response of a second order linear constant coefficient system to a sinusoidal input. We will pay special attention to the way the output changes as the frequency of the input changes. This is what we mean by the frequency response of the system. In particular, we will look at the amplitude response and the phase response; that is, the amplitude and phase lag of the system’s output considered as functions of the input frequency. In O.4 the Exponential Input Theorem was used to find a particular solution in the case of exponential or sinusoidal input. Here we will work out in detail the formulas for a second order system. We will then interpret these formulas as the frequency response of a mechanical system. In particular, we will look at damped-spring-mass systems. We will study carefully two cases: first, when the mass is driven by pushing on the spring and second, when the mass is driven by pushing on the dashpot. Both these systems have the same form p(D)x = q(t), but their amplitude responses are very different. This is because, as we will see, it can make physical sense to designate something other than q(t) as the input. For example, in the system mx0 + bx0 + kx = by0 we will consider y to be the input. (Of course, y is related to the expression on the right- hand-side of the equation, but it is not exactly the same.) 2. Sinusoidally Driven Systems: Second Order Constant Coefficient DE’s We start with the second order linear constant coefficient (CC) DE, which as we’ve seen can be interpreted as modeling a damped forced harmonic oscillator.
    [Show full text]
  • Multidisciplinary Design Project Engineering Dictionary Version 0.0.2
    Multidisciplinary Design Project Engineering Dictionary Version 0.0.2 February 15, 2006 . DRAFT Cambridge-MIT Institute Multidisciplinary Design Project This Dictionary/Glossary of Engineering terms has been compiled to compliment the work developed as part of the Multi-disciplinary Design Project (MDP), which is a programme to develop teaching material and kits to aid the running of mechtronics projects in Universities and Schools. The project is being carried out with support from the Cambridge-MIT Institute undergraduate teaching programe. For more information about the project please visit the MDP website at http://www-mdp.eng.cam.ac.uk or contact Dr. Peter Long Prof. Alex Slocum Cambridge University Engineering Department Massachusetts Institute of Technology Trumpington Street, 77 Massachusetts Ave. Cambridge. Cambridge MA 02139-4307 CB2 1PZ. USA e-mail: pjgl2@eng.cam.ac.uk e-mail: slocum@mit.edu tel: +44 (0) 1223 332779 tel: +1 617 253 0012 For information about the CMI initiative please see Cambridge-MIT Institute website :- http://www.cambridge-mit.org CMI CMI, University of Cambridge Massachusetts Institute of Technology 10 Miller’s Yard, 77 Massachusetts Ave. Mill Lane, Cambridge MA 02139-4307 Cambridge. CB2 1RQ. USA tel: +44 (0) 1223 327207 tel. +1 617 253 7732 fax: +44 (0) 1223 765891 fax. +1 617 258 8539 . DRAFT 2 CMI-MDP Programme 1 Introduction This dictionary/glossary has not been developed as a definative work but as a useful reference book for engi- neering students to search when looking for the meaning of a word/phrase. It has been compiled from a number of existing glossaries together with a number of local additions.
    [Show full text]
  • To Learn the Basic Properties of Traveling Waves. Slide 20-2 Chapter 20 Preview
    Chapter 20 Traveling Waves Chapter Goal: To learn the basic properties of traveling waves. Slide 20-2 Chapter 20 Preview Slide 20-3 Chapter 20 Preview Slide 20-5 • result from periodic disturbance • same period (frequency) as source 1 f • Longitudinal or Transverse Waves • Characterized by – amplitude (how far do the “bits” move from their equilibrium positions? Amplitude of MEDIUM) – period or frequency (how long does it take for each “bit” to go through one cycle?) – wavelength (over what distance does the cycle repeat in a freeze frame?) – wave speed (how fast is the energy transferred?) vf v Wavelength and Frequency are Inversely related: f The shorter the wavelength, the higher the frequency. The longer the wavelength, the lower the frequency. 3Hz 5Hz Spherical Waves Wave speed: Depends on Properties of the Medium: Temperature, Density, Elasticity, Tension, Relative Motion vf Transverse Wave • A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave Longitudinal Wave A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave: Pulse Tuning Fork Guitar String Types of Waves Sound String Wave PULSE: • traveling disturbance • transfers energy and momentum • no bulk motion of the medium • comes in two flavors • LONGitudinal • TRANSverse Traveling Pulse • For a pulse traveling to the right – y (x, t) = f (x – vt) • For a pulse traveling to
    [Show full text]
  • Resonance Beyond Frequency-Matching
    Resonance Beyond Frequency-Matching Zhenyu Wang (王振宇)1, Mingzhe Li (李明哲)1,2, & Ruifang Wang (王瑞方)1,2* 1 Department of Physics, Xiamen University, Xiamen 361005, China. 2 Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China. *Corresponding author. wangrf@xmu.edu.cn Resonance, defined as the oscillation of a system when the temporal frequency of an external stimulus matches a natural frequency of the system, is important in both fundamental physics and applied disciplines. However, the spatial character of oscillation is not considered in the definition of resonance. In this work, we reveal the creation of spatial resonance when the stimulus matches the space pattern of a normal mode in an oscillating system. The complete resonance, which we call multidimensional resonance, is a combination of both the spatial and the conventionally defined (temporal) resonance and can be several orders of magnitude stronger than the temporal resonance alone. We further elucidate that the spin wave produced by multidimensional resonance drives considerably faster reversal of the vortex core in a magnetic nanodisk. Our findings provide insight into the nature of wave dynamics and open the door to novel applications. I. INTRODUCTION Resonance is a universal property of oscillation in both classical and quantum physics[1,2]. Resonance occurs at a wide range of scales, from subatomic particles[2,3] to astronomical objects[4]. A thorough understanding of resonance is therefore crucial for both fundamental research[4-8] and numerous related applications[9-12]. The simplest resonance system is composed of one oscillating element, for instance, a pendulum. Such a simple system features a single inherent resonance frequency.
    [Show full text]
  • Understanding What Really Happens at Resonance
    feature article Resonance Revealed: Understanding What Really Happens at Resonance Chris White Wood RESONANCE focus on some underlying principles and use these to construct The word has various meanings in acoustics, chemistry, vector diagrams to explain the resonance phenomenon. It thus electronics, mechanics, even astronomy. But for vibration aspires to provide a more intuitive understanding. professionals, it is the definition from the field of mechanics that is of interest, and it is usually stated thus: SYSTEM BEHAVIOR Before we move on to the why and how, let us review the what— “The condition where a system or body is subjected to an that is, what happens when a cyclic force, gradually increasing oscillating force close to its natural frequency.” from zero frequency, is applied to a vibrating system. Let us consider the shaft of some rotating machine. Rotor Yet this definition seems incomplete. It really only states the balancing is always performed to within a tolerance; there condition necessary for resonance to occur—telling us nothing will always be some degree of residual unbalance, which will of the condition itself. How does a system behave at resonance, give rise to a rotating centrifugal force. Although the residual and why? Why does the behavior change as it passes through unbalance is due to a nonsymmetrical distribution of mass resonance? Why does a system even have a natural frequency? around the center of rotation, we can think of it as an equivalent Of course, we can diagnose machinery vibration resonance “heavy spot” at some point on the rotor. problems without complete answers to these questions.
    [Show full text]
  • PHASOR DIAGRAMS II Fault Analysis Ron Alexander – Bonneville Power Administration
    PHASOR DIAGRAMS II Fault Analysis Ron Alexander – Bonneville Power Administration For any technician or engineer to understand the characteristics of a power system, the use of phasors and polarity are essential. They aid in the understanding and analysis of how the power system is connected and operates both during normal (balanced) conditions, as well as fault (unbalanced) conditions. Thus, as J. Lewis Blackburn of Westinghouse stated, “a sound theoretical and practical knowledge of phasors and polarity is a fundamental and valuable resource.” C C A A B B Balanced System Unbalanced System With the proper identification of circuits and assumed direction established in a circuit diagram (with the use of polarity), the corresponding phasor diagram can be drawn from either calculated or test data. Fortunately, most relays today along with digital fault recorders supply us with recorded quantities as seen during fault conditions. This in turn allows us to create a phasor diagram in which we can visualize how the power system was affected during a fault condition. FAULTS Faults are unavoidable in the operation of a power system. Faults are caused by: • Lightning • Insulator failure • Equipment failure • Trees • Accidents • Vandalism such as gunshots • Fires • Foreign material Faults are essentially short circuits on the power system and can occur between phases and ground in virtually any combination: • One phase to ground • Two phases to ground • Three phase to ground • Phase to phase As previously instructed by Cliff Harris of Idaho Power Company: Faults come uninvited and seldom leave voluntarily. Faults cause voltage to collapse and current to increase. Fault voltage and current magnitude depend on several factors, including source strength, location of fault, type of fault, system conditions, etc.
    [Show full text]
  • Characteristics of Sine Wave Ac Power
    CHARACTERISTICS OF BY: VIJAY SHARMA SINE WAVE AC POWER: ENGINEER DEFINITIONS OF ELECTRICAL CONCEPTS, specifications & operations Excerpt from Inverter Charger Series Manual CHARACTERISTICS OF SINE WAVE AC POWER 1.0 DEFINITIONS OF ELECTRICAL CONCEPTS, SPECIFICATIONS & OPERATIONS Polar Coordinate System: It is a two-dimensional coordinate system for graphical representation in which each point on a plane is determined by the radial coordinate and the angular coordinate. The radial coordinate denotes the point’s distance from a central point known as the pole. The angular coordinate (usually denoted by Ø or O t) denotes the positive or anti-clockwise (counter-clockwise) angle required to reach the point from the polar axis. Vector: It is a varying mathematical quantity that has a magnitude and direction. The voltage and current in a sinusoidal AC voltage can be represented by the voltage and current vectors in a Polar Coordinate System of graphical representation. Phase, Ø: It is denoted by"Ø" and is equal to the angular magnitude in a Polar Coordinate System of graphical representation of vectorial quantities. It is used to denote the angular distance between the voltage and the current vectors in a sinusoidal voltage. Power Factor, (PF): It is denoted by “PF” and is equal to the Cosine function of the Phase "Ø" (denoted CosØ) between the voltage and current vectors in a sinusoidal voltage. It is also equal to the ratio of the Active Power (P) in Watts to the Apparent Power (S) in VA. The maximum value is 1. Normally it ranges from 0.6 to 0.8. Voltage (V), Volts: It is denoted by “V” and the unit is “Volts” – denoted as “V”.
    [Show full text]
  • Last Time: LRC Circuits with Phasors…
    Last time: LRC Circuits with phasors… I X where . R L εm XLL ≡ ω C L ⇒ φ IR 1 ∼ X C ≡ I XC ωC ε The phasor diagram gives us graphical solutions for φ and I: 2 2 2 2 ε=IRXXm ( +( LC − ) ) I XL I XC εm ⇓ φ 2 2 XX− I=ε m R + X()LC − X = IZ IR tanφ = LC R 2 2 ZRXX≡ +()LC − LRC series circuit; Summary of instantaneous Current and voltages VR = IR VL = IXL VC = IXC i t()= Icos (ω ) t I XL vR t()= IRcos (ω ) t ε 1 m v t()= IXcos (ω− t ) 90 = cos I ()ωt − 90 C C ωC v tL ()= IXcosL (ω+ t ) 90ω = cos I L( ω+ ) 90 t IR tε()= v t() = ω IZcos (+ φ ) t = εcos (t ω+ ) φ ad m I XC VVLC− ωLC−1 / ω 2 2 tanφ = = ZXXX=(()R + LC - ) VR R Lagging & Leading The phase φ between the current and the driving emf depends on the relative magnitudes of the inductive and capacitive reactances. XL≡ ω ε XXLC− L m tan φ = I = 1 Z R X ≡ C ω C XL Z XL XL φ Z R φ R R Z XC XC XC XL > XC XL < XC XL = XC φ > 0 φ < 0 φ = 0 current current current LAGS LEADS IN PHASE WITH applied voltage applied voltage applied voltage Lecture 19, Act 2 2A A series RC circuit is driven by emf ε. Which of the following could be an appropriate phasor ~ diagram? V ε L εm m VC VR VR VR V C εm VC (a) (b) (c) 2B For this circuit which of the following is true? (a) The drive voltage is in phase with the current.
    [Show full text]
  • EE301 – INTRO to AC and SINUSOIDS Learning Objectives
    EE301 – INTRO TO AC AND SINUSOIDS Learning Objectives a. Compare AC and DC voltage and current sources as defined by voltage polarity, current direction and magnitude over time b. Define the basic sinusoidal wave equations and waveforms, and determine amplitude, peak to peak values, phase, period, frequency, and angular velocity c. Determine the instantaneous value of a sinusoidal waveform d. Graph sinusoidal wave equations as a function of time and angular velocity using degrees and radians e. Define effective / root mean squared values f. Define phase shift and determine phase differences between same frequency waveforms Alternating Current (AC) With the exception of short-term capacitor and inductor transients, all voltages and currents we have seen up to this point have been “DC”—i.e., fixed in magnitude. Now we shift our focus to “AC” voltage and current sources. AC sources (usually represented by lowercase e(t) or i(t)) have a sinusoidal waveform. For an AC voltage, for example, the voltage polarity changes every cycle. On the other hand, for an AC current, the current changes direction each cycle with the source voltage. Voltage and Current Conventions When e has a positive value, its actual polarity is the same as the reference polarity. When i has a positive value, its actual direction is the same as the reference arrow. 1 9/14/2016 EE301 – INTRO TO AC AND SINUSOIDS Sinusoids Since our ac waveforms (voltages and currents) are sinusoidal, we need to have a ready familiarity with the equation for a sinusoid. The horizontal scale, referred to as the “time scale” can represent degrees or time.
    [Show full text]
  • Glossary Radiation Physics Phase
    Glossary Radiation Physics Phase space co-ordinates : (x,y,z,E,θ,φ), where (x,y,z) are the Cartesian coordinates of the point in configuration space where the field is evaluated, E is the particle (generically including photons) energy, and (θ,φ) are the spherical polar coordinates describing the particle direction of motion. Condensed form of phase space coordinates: (r,E) where r'xLx%yLy%zLz and E=EΩ. The vector Ω'sinθcosφLx%sinθsinφLy%cosθLz , and is a unit vector pointing in the direction of the particle velocity. Phase space volume element: d 3rd 3E'd 3rdEdΩ'dxdydzdEsinθdθdφ (cm 3&MeV&Sr) Total number of particles or photons at time t: N(t) Angular number density spectrum: d 6N n˜(r,E,t)' (cm &3&Mev &1&Sr &1) d 3rd 3E Number density spectrum: n(r,E,t)' n˜(r,E,t)dΩ m 4π Angular fluence rate spectrum: d 6N n˜ (r,E,t)'n˜(r,E,t)v' (cm &2&MeV &1&Sr &1&s &1) 3 dAzd Edt 3 where (d r'dAzvdt) Also known as angular flux density spectrum. Fluence rate spectrum: d 4N n(r,E,t)' n˜ (r,E,t)dΩ' (cm &2&MeV &1&s &1) m dAdEdt 4π Where dA is the cross sectional area of sphere centered at r Vector Fluence rate spectrum or current density: J(r,E,t)' n˜ (r,E,t)ΩdΩ (cm &2&MeV &1&s &1) m 4π d 4N J(r,E,t)@ν' Where dA is an elemental area with normal ν dAdEdt Angular fluence spectrum: t Φ˜ (r,E)' φ˜(r,E,tN)dtN (cm &2&MeV &1&Sr &1) m 0 Where t is the total exposure duration Fluence Spectrum: Φ(r,E)' Φ˜ (r,E)dΩ (cm &2&MeV &1) m 4π Energy flow vector: g(r,t)' φ˜(r,E,t)Ed 3E (MeV&cm &2&s &1) m Spectral radiance: The radiant flux per unit cross sectional area per unit solid angle per unit wave length 6 d R &2 &1 &1 Lλ(r,Ω,λ,t)' (W&m Sr nm ) dAzdΩdλ Spectral irradiance: The radiant flux per unit surface area per unit wavelength: E (r,λ,t)' L (r,Ω,λ,t)cosθdΩ (W&m &2nm &1) λ m λ Linear attenuation coefficient: The probability per unit particle path length for an interaction with the medium in which it is propagating.
    [Show full text]
  • Multiple-Wavelength Phase-Shifting Interferometry
    Multiple-wavelength phase-shifting interferometry Yeou-Yen Cheng and James C. Wyant This paper describes a method to enhance the capability of two-wavelength phase-shifting interferometry. By introducing the phase data of a third wavelength, one can measure the phase of a very steep wave front. Experiments have been performed using a linear detector array to measure surface height of an off-axis pa- rabola. For the wave front being measured the optical path difference between adjacent detector pixels was as large as 3.3 waves. After temporal averaging of five sets of data, the repeatability of the measurement is better than 25-Å rms (l = 6328 Å). I. Introduction alent wavelength l eq according to Eq, (1) with the as- Conventional single-wavelength phase-shifting in- sumption that the difference of OPD between any ad- 1-3 terferometry (PSI) is a technique which can perform jacent pixels is <l eq/2: direct phase measurement by first obtaining three or more intensity readings of fringe patterns with some known amounts of phase shifts and then using these intensity readings to calculate the phase value at each data point. Since the phase distribution across the interferogram is a measured modulo 2p one needs an Once these D OPDs between pixels were obtained, the assumption to remove the 2p discontinuities in the relative wave-front (OPD) plot or relative surface height measured phase data. The fundamental assumption plot could be obtained by integrating all these D OPD for single-wavelength PSI is that the difference of op- values. The problem for the first method of the two- tical path difference (OPD) between any adjacent pixels wavelength phase-shifting interferometer (TWLPSI) is £l/2.
    [Show full text]