DOCSLIB.ORG

Ch 1-Complex Representation of EM Waves

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ch 1-Complex Representation of EM Waves The Wave Equation and the Speed of Light Chapter 1 Physics 208, Electro-optics Peter Beyersdorf Document info Ch 1 Class Outline Maxwell’s equations Boundary conditions Poynting’s theorem and conservation laws Complex function formalism time average of sinusoidal products Wave equation Ch 1, 2 Maxwell’s Equations → → Electric field (E) and magnetic field (H) in free- space can be generalized to the electric → → displacement (D) and the magnetic induction (B) that include the effects of matter. Maxwell’s equations relate these vectors ∂B! ! E! + = 0 ∇ × ∂t Faraday’s law ∂D! ! H! = J! Ampere’s law ∇ × − ∂t ! D! = ρ Gauss’ law (for magnetism) ∇ · ! B! = 0 Gauss’ law (for electricity) ∇ · What do each of these mean? Ch 1, 3 Faraday’s Law ∂B! ! E! + = 0 ∇ × ∂t The Curl of the electric field is caused by changing magnetic fields A changing magnetic field can produce electric fields with field lines that close on themselves Ch 1, 4 Ampere’s Law ∂D! ! H! = J! ∇ × − ∂t The Curl of the magnetic field is caused by current of charged particles (J) or of the field they produce (dD/dt) A changing electric field can produce magnetic fields (with field lines that close on themselves) For all cases considered in this class, J=0 Ch 1, 5 Gauss’ Law ! D! = ρ ∇ · Electrical charges are the source of the electric field D! = "E! = "0E! + P! For all cases considered in this class, ρ=0 ε is a 3x3 tensor not a scalar (unless the material is isotropic)! ε may be a function of E and H! (giving rise to non- linear optics) ε can be determined via measurements on a parallel plate capacitor filled with a given material using the 6 equation C=εA/d Ch 1, Gauss’ Law for Magnetism There are no source of magnetic fields No magnetic monopoles Magnetic field lines can only circulate B! = µH! = µ0H! + M! μ is a 3x3 tensor not a scalar (unless the material is isotropic)! μ may be a function of E and H! (giving rise to non- linear optics) μ can be measured using the Biot-Savart law Ch 1, 7 Waves and Maxwell’s Equations A charged particle is a source of an electric field When that particle moves it changes the (spatial distribution of) the electric field When the electric field changes it produces a circulating magnetic field If the particle accelerates this circulating magnetic field will change A changing magnetic field produces a circulating electric field The circulating electric field becomes the source of a circulating magnetic field Ch 1, 8 Boundary Conditions When an EM wave propagates across an interface, Maxwell’s equations must be satisfied at the interface as well as in the bulk materials. The constraints necessary for this to occur are called the “boundary conditions” !1, µ1 !2, µ2 D! dA! = σdA · ! " d E! d!s = B! dA! · −dt · ! " B! dA! = 0 · ! d H! d!s = J! dA! + D! dA! · · dt · ! " " Ch 1, 9 Boundary Conditions Gauss’ law can be used to find the boundary conditions on the component of the electric field that is perpendicular to the interface. If the materials are dielectrics there will be no free charge on the surface (σ=0) !1, µ1 !2, µ2 D! dA! = σdA · ! " d E! d!s = B! dA! · −dt · ! " B! dA! = 0 · ! d H! d!s = J! dA! + D! dA! · · dt · ! " 0" D = D D1 D2 = σdA ∴ 1 2 Ch 1, 10 ⊥ − ⊥ ⊥ ⊥ ! → Boundary Conditions Faraday’s law can be applied at the interface. If the loop around which the electric field is computed is made to have an infintesimal area the right side will go to zero giving a relationship between the parallel components of the electric field !1, µ1 !2, µ2 D! dA! = σdA · ! " d E! d!s = B! dA! · −dt · ! " B! dA! = 0 · ! d H! d!s = J! dA! + D! dA! · · dt · ! " d " 0 E2 E1 = B dA E1 = E2 ! − ! −dt · ∴ ! ! Ch 1, 11 ! → Boundary Conditions Gauss’ law for magnetism gives a relationship between the perpendicular components of the magnetic field at the interface !1, µ1 !2, µ2 D! dA! = σdA · ! " d E! d!s = B! dA! · −dt · ! " B! dA! = 0 · ! d H! d!s = J! dA! + D! dA! · · dt · ! " " B1 A B2 A = 0 B1 = B2 ⊥ − ⊥ ∴ ⊥ ⊥ Ch 1, 12 Boundary Conditions Ampere’s law applied to a loop at the interface that has an infintesimal area gives a relationship between the parallel components of the magnetic field. (Note that in most common materials μ=μo) In the absence of currents J=0 so !1, µ1 !2, µ2 D! dA! = σdA · ! " d E! d!s = B! dA! · −dt · ! " B! dA! = 0 · ! d H! d!s = J! dA! + D! dA! · · dt · ! " 0 "d 0 ! ! ! ! H1 = H2 H1 L H2 L = J dA + D∴dA Ch 1, 13 ! − ! · dt · ! ! ! → ! Poynting’s Theorem The flow of electromagnetic energy is given by the Poynting vector S! = E! H! × which has a magnitude that is the power per unit area carried by an electromagnetic wave in the direction of S. S [W/m2]⇔E [v/m] H [A/m] Ch 1, 14 Complex-Function Formalism Steady-state sinusoidal functions of the form a(t) = A cos(ωt + α) can be treated as having a complex amplitude A = Aeiα such that the! function can be written as a(t) = Re Aeiωt ! # or in shorthand!!Ae!iω!t" where it is understood that the real part of this complex expression represents the or!iginal sinusoidal function Ch 1, 15 Phasors The complex amplitude of a sinusoidal function can be represented graphically by a point (often an arrow from the origin to a point) in the complex plane Im Im Im Re Re Re a(t) = cos ωt a(t) = sin ωt a(t) = cos (ωt) + sin (ωt) iπ/4 A = 1 A = i A = √2e− − Ch 1, 16 ! ! ! Phasors Addition of same-frequency sinusoidal functions involves factoring out the time dependance and simply adding the phasor amplitudes. Addition of difference frequency sinusoidal function is often simplified by factoring out a sinusoidal component at the average frequency. Multiplication of sinusoidal functions can not be done by multiplying phasors since Re [x] Re [y] = Re [xy] ! Ch 1, 17 Phasor Example For electric field amplitudes described by E1 = E10 cos (ω1t) and E2 = E20 cos (ω2t) Use the phasor representation to find a representation of E1+E2 as a slow modulation of a _ field at the average frequency ω=(ω1+ω2)/2 Ch 1, 18 Phasor Example 5 iω1t iω2t E1 + E2 = E10e + E20e 2.5 ∆E ∆E = E + eiω1t + E eiω2t avg 2 avg − 2 ! " ! " -4.8 -4 -3.2 -2.4 -1.6 -0.8 0 0.8 1.6 2.4 3.2 4 4.8 ∆E i(ω¯+ ∆ω )t ∆E i(ω¯ ∆ω )t = E + e 2 + E e − 2 avg 2 avg − 2 Example ! " !-2.5 " ∆ωt ∆ωt E=cos(2=πt)+3c2E os(2cos.5πt) + i∆E sin eiω¯t avg 2 2 -5 # ! " ! "$ 1 ∆ωt ∆ωt 2 = 4E2 cos2 + (∆E)2 sin2 eiω¯t+iα avg 2 2 # ! " ! "$ E + E ω1 + ω2 ∆E ∆ωt 10 20 ω¯ α arctan tan Eavg ≡ 2E 2 ≡ 2 ≡ 2 ! avg " #$ ∆E E E ∆ω ω ω Ch 1, 19 ≡ 10 − 20 ≡ 1 − 2 Time Averages Optical fields vary too fast to be directly detected, instead it is the irradiance averaged over many cycles that is detected as light. 1 T a(t)b(t) = A cos(ωt + α)B cos(ωt + β)dt ! " T !0 In terms of the phasor amplitudes this is 1 a(t)b(t) = Re AB∗ ! " 2 ! # " " Ch 1, 20 Poynting Vector Example For electric and magnetic fields given by iωt+φ E = E0e E H = 0 eiωt+φ where Z0 µ Z = η 0 377Ω 0 0 ≡ " ≈ ! 0 is the impedance of free space, what is the irradiance of the wave? How much power is measured by a detector of area A? Ch 1, 21 Poynting Vector Example For electric and magnetic fields given by iωt+φ E = E0e E H = 0 eiωt+φ where η0 µ η 0 377Ω 0 ≡ " ≈ ! 0 is the impedance of free space, what is the irradiance of the wave? 2 1 1 iφ E0 iφ E0 Savg = E! H! = Re EH∗ = Re E0e e− = × 2 2 η0 2η0 ! " # % & ' 2 This is analogous to Pavg$=V$ /2R for AC circuits 2 E0 Pavg = S!avg A! A · ≤ 2η0 Ch 1, 22 Wave Equation in Isotropic Materials Starting with Faraday’s law take the curl of both sides use vector calculus relationship to get Use Ampere’s law (in free space where J=0) and Gauss’ law (in free space where ρ=0) ! D! = 0 ∇ · in an isotropic medium Ch 1,23 Wave Equation in Crystals In an anisotropic medium does not simplify as much since ! ! ! ! ! ! ! ! does not imply but rather ! D! = ! "E! = "! E! + E! " ∇ · ∇ · ∇ · · ∇ where ∇ε$0. In this case it is usually easiest to write the wave equation as ∂2E! ^ ^ ! ! E! + µ" = 0 D=εE E^ k^ ∇ × ∇ × ∂t2 ! E ! k^ ^^ ^^ D = 0 or kxEkxE∇ · 2 kx^^ ^kxE^ ^^ !k !k E + µ"ω E = 0 kxkxE Ch 1,2 4 × × ! ! Spherical Solutions to the Wave Equation Consider solutions for E such that ∇2E and δ2E/dt2 are both proportional to E - allowing the two sides to differ only by a constant term. is one such solution in spherical coordinates. Using the relationship for the Laplacian of a spherically symmetric function: Show that !! ! given above is a solution to the wave equation Ch 1, 26 Solutions to the Wave Equation Recall the meaning of k and ω (k=2"/λ, ω=2"/T) we can express this as Since λ is the distance travelled by the wave in one cycle, and T is the time to travel one cycle, λ/T is the velocity of the wave, which can be determined from electrostatics and magnetostatics! Ch 1, 27 Solutions to the Wave Equation From our solution in free space and Gauss’ law in free space (ρ=0) We find that since!! ! ! only has a spatial dependance on r its divergence, given by must be implying !! ! ! ! ! ! meaning this must be a transverse wave (and isn’t a solution in anisotropic media) Ch 1, 28 Speed of Light In free-space where ε=ε0 and μ=μ0 the speed of light is defined to be c≡299792458 m/s.
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]
  • Slow and Stopped Light by Light-Matter Coherence Control
    Slow and stopped light by light-matter coherence control Jonas Tidström Doctoral Thesis in Photonics Stockholm, Sweden 2009 TRITA-ICT/MAP AVH Report 2009:09 KTH School of Information and ISSN 1653-7610 Communication Theory ISRN KTH/ICT-MAP/AVH-2009:09-SE SE-164 40 Kista ISBN 978-91-7415-429-0 SWEDEN Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framlägges till offentlig granskning för avläggande av filosofie doktorsexamen i fotonik, tors- dagen den 29 november 2009 kl 13:15 i sal 432, Forum, Kungl Tekniska Högskolan, Isafjordsgatan 39 Kista. c Jonas Tidström, September 2009 ° Typesset with LATEX 2ε on September 28, 2009 Universitetsservice US AB, September 2009, Abstract In this thesis we study light-matter coherence phenomena related to the interaction of a coherent laser field and the so-called Λ-system, a three-level quantum sys- tem (e.g., an atom). We observe electromagnetically induced transparency (EIT), slow and stored light in hot rubidium vapor. For example, a 6 µs Gaussian pulse 1 propagate at a velocity of 1kms− (to be compared with the normal velocity 1 ∼ of 300 000 km s− ). Dynamic changes of the control parameter allows us to slow down a pulse to a complete stop, store it for 100 µs, and then release it. During the storage time, and also during the release∼ process, some properties of the light pulse can be changed, e.g., frequency chirping of the pulse is obtained by means of Zeeman shifting the energy levels of the Λ-system. If, bichromatic continuous light fields are applied we observe overtone generation in the beating signal, and a narrow ‘dip’ in overtone generation efficiency on two-photon resonance, narrower than the ‘coherent population trapping’ transparency.
    [Show full text]
  • About the Phasor Pathways in Analogical Amplitude Modulations
    About the Phasor Pathways in Analogical Amplitude Modulations H.M. de Oliveira1, F.D. Nunes1 ABSTRACT Phasor diagrams have long been used in Physics and Engineering. In telecommunications, this is particularly useful to clarify how the modulations work. This paper addresses rotating phasor pathways derived from different standard Amplitude Modulation Systems (e.g. A3E, H3E, J3E, C3F). A cornucopia of algebraic curves is then derived assuming a single tone or a double tone modulation signal. The ratio of the frequency of the tone modulator (fm) and carrier frequency (fc) is considered in two distinct cases, namely: fm/fc<1 and fm/fc 1. The geometric figures are some sort of Lissajours figures. Different shapes appear looking like epicycloids (including cardioids), rhodonea curves, Lemniscates, folium of Descartes or Lamé curves. The role played by the modulation index is elucidated in each case. Keyterms: Phasor diagram, AM, VSB, Circular harmonics, algebraic curves, geometric figures. 1. Introduction Classical analogical modulations are a nearly exhausted subject, a century after its introduction. The first approach to the subject typically relies on the case of transmission of a single tone (for the sake of simplicity) and establishes the corresponding phasor diagram. This furnishes a straightforward interpretation, which is quite valuable, especially with regard to illustrating the effects of the modulation index, overmodulation effects, and afterward, distinctions between AM and NBFM. Often this presentation is rather naïf. Nice applets do exist to help the understanding of the phasor diagram [1- 4]. However ... Without further target, unpretentiously, we start a deeper investigating the dynamic phasor diagram with the aim of building applets or animations that illustrates the temporal variation of the magnitude of the amplitude modulated phasor.
    [Show full text]
  • Correspondence Between Phasor Transforms and Frequency Response Function in Rlc Circuits
    CORRESPONDENCE BETWEEN PHASOR TRANSFORMS AND FREQUENCY RESPONSE FUNCTION IN RLC CIRCUITS Hassan Mohamed Abdelalim Abdalla Polytechnic Department of Engineering and Architecture, University of Studies of Udine, 33100 Udine, Italy. E-mail: [email protected] Abstract: The analysis of RLC circuits is usually made by considering phasor transforms of sinusoidal signals (characterized by constant amplitude, period and phase) that allow the calculation of the AC steady state of RLC circuits by solving simple algebraic equations. In this paper I try to show that phasor representation of RLC circuits is analogue to consider the frequency response function (commonly designated by FRF) of the total impedance of the circuit. In this way I derive accurate expressions for the resonance and anti-resonance frequencies and their corresponding values of impedances of the parallel and series RLC circuits respectively, notwithstanding the presence of damping effects. Keywords: Laplace transforms, phasors, Frequency response function, RLC circuits. 1. Introduction and mathematical background RLC circuits have many applications as oscillator circuits described by a second-order differential equation. The three circuit elements, resistor R, inductor L and capacitor C can be combined in different manners. All three elements in series or in parallel are the simplest and most straightforward to analyze. RLC circuits are analyzed mathematically in the phasor domain instead of solving differential equations in the time domain. Generally, a time-dependent sinusoidal function is expressed in the following way è [1]: () (1.1) where is the amplitude, is the angular frequency and is the initial phase of . () = sin ( + ) The medium value of is given by: () () (1.2) 1 2 = |()| = where is the period of the sinusoidal function .
    [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]
  • Complex Numbers and Phasor Technique
    RaoApp-Av3.qxd 12/18/03 5:44 PM Page 779 APPENDIX A Complex Numbers and Phasor Technique In this appendix, we discuss a mathematical technique known as the phasor technique, pertinent to operations involving sinusoidally time-varying quanti- ties.The technique simplifies the solution of a differential equation in which the steady-state response for a sinusoidally time-varying excitation is to be deter- mined, by reducing the differential equation to an algebraic equation involving phasors. A phasor is a complex number or a complex variable. We first review complex numbers and associated operations. A complex number has two parts: a real part and an imaginary part. Imag- inary numbers are square roots of negative real numbers. To introduce the con- cept of an imaginary number, we define 2-1 = j (A.1a) or ;j 2 =-1 (A.1b) 1 2 Thus, j5 is the positive square root of -25, -j10 is the negative square root of -100, and so on.A complex number is written in the form a + jb, where a is the real part and b is the imaginary part. Examples are 3 + j4 -4 + j1 -2 - j2 2 - j3 A complex number is represented graphically in a complex plane by using Rectangular two orthogonal axes, corresponding to the real and imaginary parts, as shown in form Fig.A.1, in which are plotted the numbers just listed. Since the set of orthogonal axes resembles the rectangular coordinate axes, the representation a + jb is 1 2 known as the rectangular form. 779 RaoApp-Av3.qxd 12/18/03 5:44 PM Page 780 780 Appendix A Complex Numbers and Phasor Technique Imaginary (3 ϩ j4) 4 (Ϫ4 ϩ j1) 1 Ϫ2 2 Ϫ 4 0 3 Real Ϫ2 (Ϫ2 Ϫj2) Ϫ3 Ϫ FIGURE A.1 (2 j3) Graphical representation of complex numbers in rectangular form.
    [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: [email protected] e-mail: [email protected] 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. [email protected] 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]
  • Lesson 1: Representation of Ac Voltages and Currents1
    1/22/2016 Lesson 1: Phasors and Complex Arithmetic ET 332b Ac Motors, Generators and Power Systems lesson1_et332b.pptx 1 Learning Objectives After this presentation you will be able to: Write time equations that represent sinusoidal voltages and currents found in power systems. Explain the difference between peak and RMS electrical quantities. Write phasor representations of sinusoidal time equations. Perform calculations using both polar and rectangular forms of complex numbers. lesson1_et332b.pptx 2 1 1/22/2016 Ac Analysis Techniques Time function representation of ac signals Time functions give representation of sign instantaneous values v(t) Vmax sin(t v ) Voltage drops e(t) Emax sin(t e ) Source voltages Currents i(t) Imax sin(t i ) Where Vmax = maximum ( peak) value of voltage Emax = maximum (peak) value of source voltage Imax = maximum (peak) value of current v, e, i = phase shift of voltage or current = frequency in rad/sec Note: =2pf lesson1_et332b.pptx 3 Ac Signal Representations Ac power system calculations use effective values of time waveforms (RMS values) Therefore: V E I V max E max I max RMS 2 RMS 2 RMS 2 1 Where 0.707 2 So RMS quantities can be expressed as: VRMS 0.707Vmax ERMS 0.707Emax IRMS 0.707Imax lesson1_et332b.pptx 4 2 1/22/2016 Ac Signal Representations Ac power systems calculations use phasors to represent time functions Phasor use complex numbers to represent the important information from the time functions (magnitude and phase angle) in vector form. Phasor Notation I I V VRMS RMS or or I I V VRMS RMS Where: VRMS, IRMS = RMS magnitude of voltages and currents = phase shift in degrees for voltages and currents lesson1_et332b.pptx 5 Ac Signal Representations Time to phasor conversion examples, Note all signal must be the same frequency Time function-voltage Find RMS magnitude v(t) 170sin(377t 30 ) VRMS 0.707170 120.2 V Phasor V 120.230 V Time function-current Find RMS magnitude i(t) 25sin(377t 20 ) IRMS 0.70725 17.7 A Phasor I 17.7 20 A Phase shift can be given in either radians or degrees.
    [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]
  • Waveguide Propagation
    NTNU Institutt for elektronikk og telekommunikasjon Januar 2006 Waveguide propagation Helge Engan Contents 1 Introduction ........................................................................................................................ 2 2 Propagation in waveguides, general relations .................................................................... 2 2.1 TEM waves ................................................................................................................ 7 2.2 TE waves .................................................................................................................... 9 2.3 TM waves ................................................................................................................. 14 3 TE modes in metallic waveguides ................................................................................... 14 3.1 TE modes in a parallel-plate waveguide .................................................................. 14 3.1.1 Mathematical analysis ...................................................................................... 15 3.1.2 Physical interpretation ..................................................................................... 17 3.1.3 Velocities ......................................................................................................... 19 3.1.4 Fields ................................................................................................................ 21 3.2 TE modes in rectangular waveguides .....................................................................
    [Show full text]