Sinusoidal Response Notes
|
|
- Spencer Page
- 5 years ago
- Views:
Transcription
1 ECE 30 Sinusoidal Rspons Nots For BIBO Systms AStolp /29/3 Th sinusoidal rspons of a systm is th output whn th input is a sinusoidal (which starts at tim 0) Systm Sinusoidal Rspons stp input H( s) output stp rspons X m x( y( y ss ( t + t t y tr ( t Sinusoidal Input Complt stp rspons stady-stat rspons + cos( ω u( t ) <> transint rspons Y( s ) X( s) H( s ) X m H( u( t ) + Y tr ( s) H( ω ) s sin( ω u( t ) <> Gnral sinusoidal input: X mc cos( ω t ) X ms ( sin( ω u( ) u( X( s ) Stady-Stat Rspons & H( X Y( s ) X( s) mc s X ms ω H( s ) H( s ) X mc s X ms ω partial fraction xpansion: Y( s ) H( s ) Complt sinusoidal rspons A s B ω + phasor-typ transfr function C ( ) b X mc s X ms ω D ( ) E ( ) + A s B ω C D E + s ( ) ( ) ( ) stady-stat rspons + transint rspons Y ss ( s ) + Y tr ( s) multiply both sids by: X mc s X ms ω H( s ) A s B C D E ω + ( ) ( ) ( ) st s ω X mc ω X ms ω H( ω ) A ω B C D E ω + 0 ( ) ( ) ( ) X( H( ω ) A ω B ω Y ss ( s ) stady-stat rspons in phasor form X( ω ) th input xprssd in phasor form H( ω ) th stady-stat sinusoidal transfr function phasor-typ transfr function Th transint part would b found by finishing th partial-fraction xpansion Sinusoidal Rspons Nots p
2 Stady-Stat Rspons by Phasors Exprssion of signals as phasors T Priod Sinusoidal Rspons Nots p2 f frquncy, cycls / scond f T ω 2 π ω radian frquncy, radians/sc ω 2 π f A amplitud t Phas: φ T 360 or: φ t T 2 π rad y( t ) A cos( ω t φ) Phasor voltag: v( t ) V p cos( ω t φ) V( ω ) V p φ Phas: currnt: i( t ) I p cos( ω t φ) I( ω ) I p φ Ex Lt's assum th input to your systm is v ( t ) 32 cos( ω t V ( ω ) 32 or: V ( ω ) 32V / o In rctangular form: or: 32 cos( 309 V 32 sin( 0828 V V ( ω ) ( ) V Ex2 What if a signal is th sum of two sinusoids v ( t ) 32 cos( ω t v ( v 2 ( t ) 4 sin( ω t 60 v 3 ( t ) v ( v 2 ( I'm going to drop th ( notation from th phasor notation, it gts cumbrsom, but rmmbr that phasors ar in th frquncy domain From Ex: V 32 V 32V / o V V v 2 ( tim Phasors ar basd on cosins, so xprss v 2 ( as a cosin Rmmbr: sin( ωt ) cos( ω t 90 So: v 2 ( t ) 4 cos( ω t ) 4 cos( ω t 30 V 2 4V /-30 o or: V 2 4 V 30 4 cos( V 4 sin( V V V \ } V V / Add ral parts: Add imaginary parts: Chang V 3 back to polar coordinats: atan OR, in Mathcad notation (you'll s ths in futur solutions): V 3 V 3 73 V arg V 3 V 3 ( ω ) 73V /- o or: V 3 ( ω ) 73 may also b convrtd back to th tim domain: v 3 ( t ) v ( v 2 ( t ) 73 cos( ω t V Sinusoidal Rspons Nots p2 V 3 V V 2 add V V sum drawing of th phasor diagram
3 Magnitud and Phas of transfr functions With stady-stat sinusoidal inputs Ex3 a) Find th magnitud and phas of th following transfr function at this frquncy: ω 2 rad sc H( s ) s ω H( ω ) 2 s 2 s 20 s 2 s 0 2 ( 2 ( 20 ( 2 sc ( ω ) 2 s 2 sc s 20 sc 2 s 2 sc s 0 sc 2 0 sc 2 Exprssd with propr units 20 2 ω 2 ( 0 ω 2 ( without units ( 2) ( 2) H( ω ) M / H( atan 0 atan b) Find th stady-stat sinusoidal output if th input is: 32 cos( 2 t V in 32 V 32V / o V outss ( ω ) V in ( H( ω ) 32 V v outss ( t ) 74 cos( 2 t 3637 ( V Ex4 a) Find th magnitud and phas of th following transfr function at this frquncy: f Hz ω 2 π f H( s ) s 2 20 sc s 000 sc 2 s 2 0 sc s 800 sc 2 s ω 342 rad sc H( ω ) ( 2 20 sc ( ω ) ( 2 0 sc ( ω ) 000 sc sc 2 ( 342 ) 20 ( 342) 000 ( 342) 2 0 ( 342) ω 342 rad sc without units H( ω ) M / H( atan b) Find th stady-stat sinusoidal output if th input is: x( t ) 4 cos( 2 π Hz X( ω ) 4 0 and thn Y( ω ) 4 ( ) Not that you can us th rctangular form of H( y( t ) 836 cos 342 rad sc t 364 sin 342 rad sc t not that th sin carris th opposit sign as th imaginary part atan y( t ) 688 cos 342 rad sc t 32 Sinusoidal Rspons Nots p3
4 Impdancs sris: Z q Z Z 2 Z 3 + Exampl: Z q R C s L s Voltag dividr: V Zn V total Z n Z Z 2 Z 3 + paralll: Exampl: Z q + Z Z 2 Z 3 Currnt dividr: Z q R C s L s I Zn I total R C s Z n + Z Z 2 Z 3 L s Ex4 a) Find th stady-stat V R and v R ( givn v S ( is a 2 Vpp cosin wav at: f 2 khz b) Find th currnt: I 6 R L ( R 00 Ω V S ( ω ) 6 0 ω 2 π f ω L 80 mh Transfr function for V R as th output: H( s ) C 04 µf 0 I( s) V( s) C ( V R ( s) V S ( s) 266 rad sc R R L s V R ( ω ) R 6 V V V R 363V /-82 o R L ( C ( v R ( t ) 363 cos 266 rad sc t 82 Z( s) I( s ) V( s) Z( s) 6 V ( 266) s ω 266 rad sc ( 266 ) 6 V magnitud: 6324 ma 9488 Ω angl: I 6324mA /-82 o 6 V C s 6 V atan c) Draw a phasor diagram of all th voltags V L I Z L 6324 ma 00 Ω 636 V V L 636V / 38 o V C I Z C 6324 ma ( 99) Ω 28 V 82 ( 90) 38 V C -28V / 38 o 28V /-482 o Sinusoidal Rspons Nots p4
5 Sinusoidal Rspons Nots p L 2 mh Ex a) Find th stady-stat V C and v C ( givn v in ( is a 2 Vp cosin wav at: with a 20 o lading phas angl Transfr function for V C as th output: H( s ) H( s ) L s R L 2 s R L 2 s C s C s R L 2 s R L 2 s f 2 khz V C ( s) C s C s /\ V in C µf R 200 Ω L 2 8 mh L s R L 2 s C s V in 2 V 20 ω 2 π f ω 708 rad sc H( ω ) L ( R L 2 ( C ( 0002 ( 708 ) ( 708) 0 6 ( 708) ( 00708) V C V in ( H( ω ) 20 2 V / o 204V / 896 o v c ( t ) 204 cos 708 rad sc t 896 a) Find th stady-stat I L2 and i L2 ( V C ( s) Transfr function for I L2 as th output: H( s ) I L2 ( s) R L 2 s V C ( s) R L 2 s H( s ) L s C s R L 2 s R L 2 s L s L s C s R L 2 s R L 2 s H( ω ) L ( ω ) L C ( 2 R L 2 ( R L 2 ( Ω 798 kω 438 I L2 V in ( H( ω ) kω kω / o 8638mA / -238 o i L2 ( t ) 8638 ma cos 708 rad sc t 238 Sinusoidal Rspons Nots p
6 Sinusoidal Rspons Nots p6 Ex6 This systm: H( s ) s 20 s Has this input: x( t ) 4 sin( 2 t 40 u( a) Us stady-stat AC analysis to find th stady-stat output y ss ( t )? AC stady-stat H( H( 2 ) atan 2 20 atan / 3646 X( ω ) 4 30 Not 90 o phas-lag bcaus it's givn as a sin wav Y ss ( ω ) / / 6646 y ss ( t ) 776 cos( 2 t 6646 b) Exprss th output, and sparat into 3 partial fractions that you can find in th laplac transform tabl without using complx numbrs Show what thy ar, but don't find th cofficints Find th input as a sum of a pur sin and cosin 4 sin( cos( so: 4 sin( 2 t 40 u( t ) ( 3064 sin( 2 t ) 27 cos( 2 ) u( Y( s ) s s 2 44 ( s 20) ( s ) A s + B s s C 2 s 2 44 c) Continu with th partial fraction xpansion ust far nough to find th transint cofficint as a numbr ( s )( s 20 ) A s B s ( s ) + C 2 ( s ) lt s ( ( ) )( 20 ) A A ( s 20 )( s) 2 44 A 4404 d) Exprss th complt (both transint and stady-sta output as a function of tim y( t )? 776 cos( sin( y( t ) 4404 t 776 cos( 2 t 6646 u( Eithr answr y( t ) 4404 t 697 cos( 2 t ) 68 sin( 2 u( ) What is th tim constant of th transint part this xprssion? τ? Sinusoidal Rspons Nots p6
ECE 2210 / 00 Phasor Examples
EE 0 / 00 Phasor Exampls. Add th sinusoidal voltags v ( t ) 4.5. cos( t 30. and v ( t ) 3.. cos( t 5. v ( t) using phasor notation, draw a phasor diagram of th thr phasors, thn convrt back to tim domain
More informationDSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer
DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016 LECTURE
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationAnnounce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review
ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More informationSAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS
It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationChapter 10: Sinusoidal Steady-State Analysis
Chapter 10: Sinusoidal Steady-State Analysis 1 Objectives : sinusoidal functions Impedance use phasors to determine the forced response of a circuit subjected to sinusoidal excitation Apply techniques
More informationCSE 245: Computer Aided Circuit Simulation and Verification
CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy
More informationRevision of Basic A.C. Theory
Revision of Basic A.C. Theory 1 Revision of Basic AC Theory (Much of this material has come from Electrical & Electronic Principles & Technology by John Bird) Electricity is generated in power stations
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More informationECE Circuit Theory. Final Examination. December 5, 2008
ECE 212 H1F Pg 1 of 12 ECE 212 - Circuit Theory Final Examination December 5, 2008 1. Policy: closed book, calculators allowed. Show all work. 2. Work in the provided space. 3. The exam has 3 problems
More informationFrequency Response. Re ve jφ e jωt ( ) where v is the amplitude and φ is the phase of the sinusoidal signal v(t). ve jφ
27 Frequency Response Before starting, review phasor analysis, Bode plots... Key concept: small-signal models for amplifiers are linear and therefore, cosines and sines are solutions of the linear differential
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More informationLecture 2: Discrete-Time Signals & Systems. Reza Mohammadkhani, Digital Signal Processing, 2015 University of Kurdistan eng.uok.ac.
Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1 Signal Dfinition and Exampls 2 Signal: any physical quantity that
More informationMATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 3 PERIODIC FUNCTIONS
MATHEMATICS FOR ENGINEERING TRIGONOMETRY TUTORIAL 3 PERIODIC FUNCTIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.
More information0WAVE PROPAGATION IN MATERIAL SPACE
0WAVE PROPAGATION IN MATERIAL SPACE All forms of EM nrgy shar thr fundamntal charactristics: 1) thy all tral at high locity 2) In traling, thy assum th proprtis of was 3) Thy radiat outward from a sourc
More informationCHAPTER 10. Consider the transmission lines for voltage and current as developed in Chapter 9 from the distributed equivalent circuit shown below.
CHAPTER 1 1. Sinusoidal Stady Stat in Transmission ins 1.1 Phasor Rprsntation of olta and Currnt Wavs Considr th transmission lins for volta and currnt as dvlopd in Chaptr 9 from th distributd quivalnt
More information1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More information22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.
Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M
More informationThings I Should Know Before I Get to Calculus Class
Things I Should Know Bfor I Gt to Calculus Class Quadratic Formula = b± b 4ac a sin + cos = + tan = sc + cot = csc sin( ± y ) = sin cos y ± cos sin y cos( + y ) = cos cos y sin sin y cos( y ) = cos cos
More informationECE 344 Microwave Fundamentals
ECE 44 Microwav Fundamntals Lctur 08: Powr Dividrs and Couplrs Part Prpard By Dr. hrif Hkal 4/0/08 Microwav Dvics 4/0/08 Microwav Dvics 4/0/08 Powr Dividrs and Couplrs Powr dividrs, combinrs and dirctional
More informationIntroduction to the quantum theory of matter and Schrödinger s equation
Introduction to th quantum thory of mattr and Schrödingr s quation Th quantum thory of mattr assums that mattr has two naturs: a particl natur and a wa natur. Th particl natur is dscribd by classical physics
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS
EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 3 TUTORIAL 1 - TRIGONOMETRICAL GRAPHS CONTENTS 3 Be able to understand how to manipulate trigonometric expressions and apply
More informationChapter 3: Capacitors, Inductors, and Complex Impedance
haptr 3: apacitors, Inductors, and omplx Impdanc In this chaptr w introduc th concpt of complx rsistanc, or impdanc, by studying two ractiv circuit lmnts, th capacitor and th inductor. W will study capacitors
More informationPhysics 160 Lecture 3. R. Johnson April 6, 2015
Physics 6 Lcur 3 R. Johnson April 6, 5 RC Circui (Low-Pass Filr This is h sam RC circui w lookd a arlir h im doma, bu hr w ar rsd h frquncy rspons. So w pu a s wav sad of a sp funcion. whr R C RC Complx
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by:
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More information2. Finite Impulse Response Filters (FIR)
.. Mthos for FIR filtrs implmntation. Finit Impuls Rspons Filtrs (FIR. Th winow mtho.. Frquncy charactristic uniform sampling. 3. Maximum rror minimizing. 4. Last-squars rror minimizing.. Mthos for FIR
More informationSlide 1. Slide 2. Slide 3 DIGITAL SIGNAL PROCESSING CLASSIFICATION OF SIGNALS
Slid DIGITAL SIGAL PROCESSIG UIT I DISCRETE TIME SIGALS AD SYSTEM Slid Rviw of discrt-tim signals & systms Signal:- A signal is dfind as any physical quantity that varis with tim, spac or any othr indpndnt
More informationChapter 2 Linear Waveshaping: High-pass Circuits
Puls and Digital Circuits nkata Ra K., Rama Sudha K. and Manmadha Ra G. Chaptr 2 Linar Wavshaping: High-pass Circuits. A ramp shwn in Fig.2p. is applid t a high-pass circuit. Draw t scal th utput wavfrm
More informationComplex representation of continuous-time periodic signals
. Complx rprsntation of continuous-tim priodic signals Eulr s quation jwt cost jsint his is th famous Eulr s quation. Brtrand Russll and Richard Fynman both gav this quation plntiful prais with words such
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationElectromagnetism Physics 15b
lctromagntism Physics 15b Lctur #8 lctric Currnts Purcll 4.1 4.3 Today s Goals Dfin lctric currnt I Rat of lctric charg flow Also dfin lctric currnt dnsity J Charg consrvation in a formula Ohm s Law vryon
More informationSINUSOIDAL STEADY STATE CIRCUIT ANALYSIS
SINUSOIDAL STEADY STATE CIRCUIT ANALYSIS 1. Introduction A sinusoidal current has the following form: where I m is the amplitude value; ω=2 πf is the angular frequency; φ is the phase shift. i (t )=I m.sin
More informationSinusoidal Response of RLC Circuits
Sinusoidal Response of RLC Circuits Series RL circuit Series RC circuit Series RLC circuit Parallel RL circuit Parallel RC circuit R-L Series Circuit R-L Series Circuit R-L Series Circuit Instantaneous
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationUsing Complex Numbers in Circuit Analysis Review of the Algebra of Complex Numbers
Physics 6.P. Johnson Spring 6 Using omplx Numbrs in ircuit Analysis viw of th Algbra of omplx Numbrs omplx numbrs ar commonly usd in lctrical nginring, as wll as in physics. In gnral thy ar usd whn som
More informationHandout 11: AC circuit. AC generator
Handout : AC circuit AC generator Figure compares the voltage across the directcurrent (DC) generator and that across the alternatingcurrent (AC) generator For DC generator, the voltage is constant For
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationExam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.
Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r
More informationREACTANCE. By: Enzo Paterno Date: 03/2013
REACTANCE REACTANCE By: Enzo Paterno Date: 03/2013 5/2007 Enzo Paterno 1 RESISTANCE - R i R (t R A resistor for all practical purposes is unaffected by the frequency of the applied sinusoidal voltage or
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationSingle Phase Parallel AC Circuits
Single Phase Parallel AC Circuits 1 Single Phase Parallel A.C. Circuits (Much of this material has come from Electrical & Electronic Principles & Technology by John Bird) n parallel a.c. circuits similar
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationOn the Hamiltonian of a Multi-Electron Atom
On th Hamiltonian of a Multi-Elctron Atom Austn Gronr Drxl Univrsity Philadlphia, PA Octobr 29, 2010 1 Introduction In this papr, w will xhibit th procss of achiving th Hamiltonian for an lctron gas. Making
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationChapter 3: Capacitors, Inductors, and Complex Impedance
haptr 3: apacitors, Inductors, and omplx Impdanc In this chaptr w introduc th concpt of complx rsistanc, or impdanc, by studying two ractiv circuit lmnts, th capacitor and th inductor. W will study capacitors
More informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 18 121025 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review RMS Values Complex Numbers Phasors Complex Impedance Circuit Analysis
More information1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:
Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any
More informationREADING ASSIGNMENTS. Signal Processing First. Problem Solving Skills LECTURE OBJECTIVES. x(t) = cos(αt 2 ) Fourier Series ANALYSIS.
Signal Procssing First Lctur 5 Priodic Signals, Harmonics & im-varying Sinusoids READING ASSIGNMENS his Lctur: Chaptr 3, Sctions 3- and 3-3 Chaptr 3, Sctions 3-7 and 3-8 Nxt Lctur: Fourir Sris ANALYSIS
More informationChapter 10: Sinusoids and Phasors
Chapter 10: Sinusoids and Phasors 1. Motivation 2. Sinusoid Features 3. Phasors 4. Phasor Relationships for Circuit Elements 5. Impedance and Admittance 6. Kirchhoff s Laws in the Frequency Domain 7. Impedance
More informationThe Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function
A gnraliation of th frquncy rsons function Th convolution sum scrition of an LTI iscrt-tim systm with an imuls rsons h[n] is givn by h y [ n] [ ] x[ n ] Taing th -transforms of both sis w gt n n h n n
More informationFinite element discretization of Laplace and Poisson equations
Finit lmnt discrtization of Laplac and Poisson quations Yashwanth Tummala Tutor: Prof S.Mittal 1 Outlin Finit Elmnt Mthod for 1D Introduction to Poisson s and Laplac s Equations Finit Elmnt Mthod for 2D-Discrtization
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationAs the matrix of operator B is Hermitian so its eigenvalues must be real. It only remains to diagonalize the minor M 11 of matrix B.
7636S ADVANCED QUANTUM MECHANICS Solutions Spring. Considr a thr dimnsional kt spac. If a crtain st of orthonormal kts, say, and 3 ar usd as th bas kts, thn th oprators A and B ar rprsntd by a b A a and
More informationAC analysis - many examples
AC analysis - many examples The basic method for AC analysis:. epresent the AC sources as complex numbers: ( ). Convert resistors, capacitors, and inductors into their respective impedances: resistor Z
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationModule 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Single-phase circuits ersion EE T, Kharagpur esson 6 Solution of urrent in Parallel and Seriesparallel ircuits ersion EE T, Kharagpur n the last lesson, the following points were described:. How
More informationPrelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours
Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd - hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn
More informationReview Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )
Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division
More informationCoupled Pendulums. Two normal modes.
Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron
More informationmywbut.com Lesson 16 Solution of Current in AC Parallel and Seriesparallel
esson 6 Solution of urrent in Parallel and Seriesparallel ircuits n the last lesson, the following points were described:. How to compute the total impedance/admittance in series/parallel circuits?. How
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationDesign Guidelines for Quartz Crystal Oscillators. R 1 Motional Resistance L 1 Motional Inductance C 1 Motional Capacitance C 0 Shunt Capacitance
TECHNICAL NTE 30 Dsign Guidlins for Quartz Crystal scillators Introduction A CMS Pirc oscillator circuit is wll known and is widly usd for its xcllnt frquncy stability and th wid rang of frquncis ovr which
More informationProblem Set #2 Due: Friday April 20, 2018 at 5 PM.
1 EE102B Spring 2018 Signal Procssing and Linar Systms II Goldsmith Problm St #2 Du: Friday April 20, 2018 at 5 PM. 1. Non-idal sampling and rcovry of idal sampls by discrt-tim filtring 30 pts) Considr
More information3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d www.clas.ucsb.du/staff/vinc
More informationAC analysis. EE 201 AC analysis 1
AC analysis Now we turn to circuits with sinusoidal sources. Earlier, we had a brief look at sinusoids, but now we will add in capacitors and inductors, making the story much more interesting. What are
More informationName (print): Lab (circle): W8 Th8 Th11 Th2 F8. θ (radians) θ (degrees) cos θ sin θ π/ /2 1/2 π/4 45 2/2 2/2 π/3 60 1/2 3/2 π/
Name (print): Lab (circle): W8 Th8 Th11 Th2 F8 Trigonometric Identities ( cos(θ) = cos(θ) sin(θ) = sin(θ) sin(θ) = cos θ π ) 2 Cosines and Sines of common angles Euler s Formula θ (radians) θ (degrees)
More informationInstructions for Section 1
Instructions for Sction 1 Choos th rspons tht is corrct for th qustion. A corrct nswr scors 1, n incorrct nswr scors 0. Mrks will not b dductd for incorrct nswrs. You should ttmpt vry qustion. No mrks
More informationDISCRETE TIME FOURIER TRANSFORM (DTFT)
DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More information( ) = ( ) ( ) ( ) ( ) τ τ. This is a more complete version of the solutions for assignment 2 courtesy of the course TA
This is a mor complt vrsion o th solutions or assignmnt courtsy o th cours TA ) Find th Fourir transorms o th signals shown in Figur (a-b). -a) -b) From igur -a, w hav: g t 4 0 t = t 0 othrwis = j G g
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationSolutions to ECE 2026 Problem Set #1. 2.5e j0.46
Solutions to ECE 2026 Problem Set #1 PROBLEM 1.1.* Convert the following to polar form: (a) z 3 + 4j 3 2 + 4 2 e jtan 1 ( 4/3) 5e j0.295 3 4j 5e (b) z ---------------- ------------------------ j0.295 3
More informationSinusoidal steady-state analysis
Sinusoidal steady-state analysis From our previous efforts with AC circuits, some patterns in the analysis started to appear. 1. In each case, the steady-state voltages or currents created in response
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts
More informationProf. Shayla Sawyer CP08 solution
What does the time constant represent in an exponential function? How do you define a sinusoid? What is impedance? How is a capacitor affected by an input signal that changes over time? How is an inductor
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More informationChapter 33. Alternating Current Circuits
Chapter 33 Alternating Current Circuits 1 Capacitor Resistor + Q = C V = I R R I + + Inductance d I Vab = L dt AC power source The AC power source provides an alternative voltage, Notation - Lower case
More informationLecture 3: Phasor notation, Transfer Functions. Context
EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of
More informationStep Response Analysis. Frequency Response, Relation Between Model Descriptions
Step Response Analysis. Frequency Response, Relation Between Model Descriptions Automatic Control, Basic Course, Lecture 3 November 9, 27 Lund University, Department of Automatic Control Content. Step
More information2. Laser physics - basics
. Lasr physics - basics Spontanous and stimulatd procsss Einstin A and B cofficints Rat quation analysis Gain saturation What is a lasr? LASER: Light Amplification by Stimulatd Emission of Radiation "light"
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More informationResidence Times Difference (RTD) - Fluxgate Magnetometer A.A. 2007/2008
Univrsità dgli Studi di atania Facoltà di Inggnria Dipartimnto di Inggnria Elttrica Elttronica di Sistmi Rsidnc Tims Diffrnc (RTD) - Fluxgat Magntomtr Ing.. arlo Trigona A.A. 007/ Outlin lassification
More informationthe output is Thus, the output lags in phase by θ( ωo) radians Rewriting the above equation we get
Th output y[ of a frquncy-sctiv LTI iscrt-tim systm with a frquncy rspons H ( xhibits som ay rativ to th input caus by th nonro phas rspons θ( ω arg{ H ( } of th systm For an input A cos( ωo n + φ, < n
More informationTypes of Transfer Functions. Types of Transfer Functions. Types of Transfer Functions. Ideal Filters. Ideal Filters
Typs of Transfr Typs of Transfr x[n] X( LTI h[n] H( y[n] Y( y [ n] h[ k] x[ n k] k Y ( H ( X ( Th tim-domain classification of an LTI digital transfr function is basd on th lngth of its impuls rspons h[n]:
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationCircuit Analysis-III. Circuit Analysis-II Lecture # 3 Friday 06 th April, 18
Circuit Analysis-III Sinusoids Example #1 ü Find the amplitude, phase, period and frequency of the sinusoid: v (t ) =12cos(50t +10 ) Signal Conversion ü From sine to cosine and vice versa. ü sin (A ± B)
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationSinusoidal Steady State Analysis
Sinusoidal Steady State Analysis 9 Assessment Problems AP 9. [a] V = 70/ 40 V [b] 0 sin(000t +20 ) = 0 cos(000t 70 ).. I = 0/ 70 A [c] I =5/36.87 + 0/ 53.3 =4+j3+6 j8 =0 j5 =.8/ 26.57 A [d] sin(20,000πt
More informationALTERNATING CURRENT. with X C = 0.34 A. SET UP: The specified value is the root-mean-square current; I. EXECUTE: (a) V = (0.34 A) = 0.12 A.
ATENATING UENT 3 3 IDENTIFY: i Icosωt and I I/ SET UP: The specified value is the root-mean-square current; I 34 A EXEUTE: (a) I 34 A (b) I I (34 A) 48 A (c) Since the current is positive half of the time
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More information6.1 Introduction
6. Introduction A.C Circuits made up of resistors, inductors and capacitors are said to be resonant circuits when the current drawn from the supply is in phase with the impressed sinusoidal voltage. Then.
More informationIf we integrate the given modulating signal, m(t), we arrive at the following FM signal:
Part b If w intgrat th givn odulating signal, (, w arriv at th following signal: ( Acos( πf t + β sin( πf W can us anothr trigonotric idntity hr. ( Acos( β sin( πf cos( πf Asin( β sin( πf sin( πf Now,
More information