FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures"

Transcription

1 FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in many areas o engineering, and mos o you will discuss he mehod again in your second year mahemaics unis We consider here Fourier series expansions o periodic uncions, ie uncions which repea hemselves exacly a regular inervals wo examples are shown below Figure (a Figure (b De A uncion is periodic o period ( > i and only i ( + =( or all hereore he period is deined as he ime inerval required or one complee lucuaion Hence ( =cos is periodic wih period π since ( +π=cos(+π=cos=( or all NB I is periodic wih period, hen clearly rom he graphs, or rom repeaed use o he deiniion, is also periodic wih periods,3, you should choose he minimum period o he uncion o be is period Ex Deermine wheher he ollowing uncions are periodic and, i so, deermine he periods:- (i ( = sin(, (ii ( =cos( 3, (iii ( =cos+ sin( (i ( = sin( is periodic wih period π since ( + π =sin(+π=sin(+π=sin=( or all (ii ( =cos( 3 is periodic wih period π/ 3 since ( + π ( ( 3 =cos + π =cos( 3+π=cos( 3=( or all 3 3

2 (iii (more complicaed cos has periods π, 4π, 6π, sin has periods π, π, 3π, and clearly he minimum period or he sum o hese quaniies is he smalles number ha appears in boh liss o periods In his example clearly =π he uncion ( =cos+cos( would be even more diicul, since cos has periods π, 4π, 6π, cos( π has periods, 4π, 6π, Since he muliplicaive acors o π in he periods or cos are whole numbers bu he corresponding acors or cos( are irraional (always involving here is no number ha appears in boh liss Hence he uncion ( =cos+cos( is NO periodic (i never repeas isel, despie is comparaively simple orm More generally, we can say ha he sum o wo or more cosine waves will be periodic only when he raios o all pairs o periods orm raional numbers (raios o inegers Sine wave Le us recall deiniions linked o a sine wave sin φ φ/ω Figure Consider ( = sin(ω + φ = sin(πν + φ, shown above, where is ampliude, ω is he circular (or angular requency in radians/uni ime, ν is requency in cycles/uni ime (Herz φ is phase angle wih respec o ime origin in radians he period o above sine wave is /ν =π/ω seconds A posiive phase angle φ shis waveorm o he le (a lead and a negaive phase angle moves waveorm o he righ (a lag π/ω Whole range Fourier series Fourier analysis decomposes a complicaed periodic wave shape ino a sum o sine and cosine waves Suppose ( = ( ( ] πn πn a + a n cos + b n sin, ( n= where ( is a periodic uncion o period and a n and b n are he (consan Fourier coeiciens o (, deined by a n = +/ ( πn (cos d, n =,,, ( /

3 b n = +/ / ( πn (sin d, n =,,3 (3 Equaions (, ( and (3 appear on he Formula Shee (NB a and a n (or n> are deined by he same inegral (, bu a mus always be calculaed separaely since orm o inegral changes when n = Equaions ( (3 also appear in he Daa Book bu in a slighly dieren orm since ha book assumes he uncion has period L (ge amiliar wih he equaions you mus use Is he orm ( reasonable? Does he ininie sum converge? Well, given a periodic uncion ( i we calculae is Fourier coeiciens using he inegrals ( and (3, and hen wrie ( a + n= a n cos ( πn + b n sin ( ] πn, (4 where he series on he RHS o (4 is called he whole-range Fourier series o (, i can be proved ha he ininie series in (4 does converge, under cerain condiions, or a wide variey o uncions Suicien condiions or convergence, known as he Dirichle condiions, are saed below I ( is a periodic uncion wih period and (i is piecewise coninuous in he inerval / <<+/ (ie i is bounded in he inerval and is coninuous in he inerval excep a a inie number o poins, (ii has only a inie number o maxima and minima in he inerval, ] hen he Fourier series converges a a poin o he value lim ( + lim ( + he value wrien above represens he average o he le and righ hand limis o a (see Figure 3 Obviously, a all poins a which he uncion ( is coninuous, he le and righ-hand limis o are he same, so he Fourier series converges o ( value o Fourier series a Figure 3 NB he symbol is used in equaion (4 o show ha ( is no necessarily equal o he series on he righ (as above discussion shows Ex Find he Fourier series o he uncion deined by {, ( = < +, < (+=( 3

4 For his uncion (shown in Figure 4 Figure 4 a = +/ / = ] a n = = +/ / (cos(d = + ] = ( +/ / ( (d = + ( { πn (cos d = { sin(πn/ (πn/ ] + { ( ( cos / } ] sin(πn/ (πn/ ( d + / =; ( πn d + +/ / } (+ d = { sin ( sin(n + sin(πn sin } = ( sin(πn + sin(πn = ; πn πn b n = +/ ( { πn (sin d = ( πn ( sin d + / / { cos(πn/ = ] + cos(πn/ ] } (πn/ (πn/ / ( } πn (+ cos d ( } πn (+ sin d = {cos cos(n cos(πn+cos}= { cos(πn} = πn πn πn { ( n } In he calculaion o b n he resul cos(nπ =( n has been used, since cos(nπ alernaes beween and + or all ineger n When n is even, ( n = + and hence b n = When n is odd hen ( n =, and b n = 4 ( ( = nπ nπ hus b = 4 π, b =, b 3 = 4 3π, b 4 =, and i ω =π/ we obain ( 4 π sin(ω + 3 sin(3ω + ] 5 sin(5ω + 4

5 Parial sums De he mh parial sum S m ( is he sum o a Fourier series up o he erms in sin(πm/ and cos(πm/ m For he Fourier series in Ex i ollows ha S m ( = b n sin(nω soha n= S (= 4 π sin ω S 3 (= 4 sin ω + ] π 3 sin 3ω S 5 ( = 4 sin ω + π 3 sin 3ω + ] 5 sin 5ω A =, a poin o disconinuiy o he original uncion, all erms in he calculaed Fourier series are zero so he oal sum is also zero his illusraes he resul ha he Fourier series is convergen a =o ] lim ( + lim ( = + ( ] = + 3 Even and odd uncions In Ex he Fourier coeiciens were calculaed direcly rom he inegral deiniions In ha example, and many ohers, calculaions can be simpliied by using properies o he uncion ( Recall a uncion ( iseven i ( =(; a uncion ( isodd i ( = ( For example, ( =cosis even since ( =cos( =cos=(, ( =sinis odd since ( = sin( = sin = ( Noe ha he produc o wo even uncions and g is even, since ( g( =(g(; he produc o wo odd uncions and g is even, since ( g( =( ( ( g( = ( g( bu he produc o an odd uncion wih an even uncion g is odd, since ( g( =( ( g( = (g( he crucial resuls concern he inegrals o odd and even uncions From he deiniions o hese uncions, or rom heir graphs, i is easily seen ha i ( isodd hen i ( iseven hen a a a a ( d =; ( d = hese inegral resuls are illusraed below: a (d a A A a A a A a Figure 5(a Figure 5(b 5

6 Figure 4 shows ha he uncion in Ex is odd, and so use o he above inegral resuls would have simpliied he calculaion o he a n and b n, in an analogous way o heir use in he example below Ex 3 Deermine he Fourier series expansion o he uncion ( deined by: i </, ( = + i / <π/, and ( +π=( or all i π/ <π, ( is shown on Figure 6 clearly uncion is even wih period =π(or L =π, i you are using he Daa Book 3π Figure 6 π π 3π Using he general orm or he Fourier coeiciens, wih =π(or L = π, b n = π +π ( nπ (sin d = π π +π ( sin(n d Since ( is even and sin(n is odd, he produc o hese uncions is odd and hence he inegral is zero, ie b n = For he coeicien a, a = +π ( cos(d = +π ( d, π π and since ( is even i is possible o rewrie he laer o give urning o he coeiciens a n, a = { +π ( d = +π/ π π = { } ] π/ + ] π π/ = π π a n = π +π (cos } π d + ( d π/ { π + ( ( nπ d = π π +π ( π } = (cos(n d Boh ( and cos(n are even, so he produc is even and hence a n = { +π (cos(n d = +π/ } π cos(n d + ( cos(n d π π π/ { sin = ] π/ ] } π n sin n π n n π/ = { ( nπ ( nπ } sin sin sin(nπ+sin = 4 ( nπ πn πn sin 6

7 Now ( nπ sin = { i n is even, + i n =,5,9, i n =3,7,, so i n is even, 4 a n = πn i n =,5,9, 4 πn i n =3,7,, and, combining ogeher hese resuls, gives he inal soluion ( ( nπ a n cos = {cos cos(3+ cos(5 } π π n= Graphs o sums o various numbers o non-zero erms in his soluion (ie he parial sums S n are shown on he nex page he igures show ha he series becomes closer o he given uncion ( as more erms are included, bu here will always remain considerable error in he viciniy o a disconinuiy no maer how many erms you use In rying o achieve he ininie slope a he disconinuiy, he parial sum acually overshoos hese addiional peaks become o vanishingly small widh as he number o erms increases (see he igures, bu he overshoo isel does no reduce o zero his resul is known as Gibbs phenomenon Inhecaseo he square wave in Ex 3 he smalles magniude o he overshoo is abou 9% Jusiicaion o ormulae in Fourier series Finally we jusiy he Fourier series ormulae he proos involve orhogonaliy relaions which will now be saed and proved: / ( ( πm πn Resul cos sin d =, or all m and n (5 / ( ( πm πn (5 holds because cos is even, sin is odd, hence he produc is odd and he above inegral is zero / ( ( πm πn Resul cos cos d =, or all m n (6 / On he LHS o (6 boh cosine uncions are even, so he produc is even he inegral on he LHS can hereore be expressed / ( ( πm πn cos cos d On using he rigonomeric ideniies on he Formula shee, or in he Daa Book, he above inegral becomes / ( ( πm πn / { ( ( } π(m + n π(m n cos cos d = cos +cos d ] / sin(π(m + n/ sin(π(m n/ = + π(m + n/ π(m n/ =, since sin(m nπ = sin(m + nπ = Using a very similar mehod (which is no given here i can be shown ha Resul 3 / / ( πm sin sin ( πn d =, or all m n (7 7

8 Figures or his page o come 8

9 he case m = n or he inegrals in (6 and (7 need o be reaed separaely: Resul 4 / / ( πm cos d =, m (8 o prove he laer / / ( πm cos d = = / + ( πm cos d = sin (4πm/ 4πm/ Using an analogous argumen i can be shown ha Resul 5 / / ] / / d =, ( +cos ( 4πm d since sin πm =sin= ( πm sin d =, m (9 he relaions (5 o (9 can be used o jusiy he expressions or he Fourier coeiciens as ollows he general ormulae or Fourier series were saed earlier, equaions ( o (4 Muliply boh sides o ( by he quaniy cos(πm/ and inegrae rom /o /ogive / ( πm (cos d = / ( a πm / / cos d { / ( ( }] πm πn + a n cos cos d n= / { / ( ( }] πm πn + b n cos sin d n= / ( Relaion (5 implies ha he hird inegral on he RHS o ( is always zero When m = hen he irs inegral on RHS o ( is, bu he second inegral = / Hence a = (d / When m, he irs inegral on RHS o ( is zero bu, using (6 and (8, he second inegral is non-zero only when m = n I ollows ha a m = / ( πm (cos d / In a similar way, on muliplying ( by sin(πm/ and inegraing, we can show b m = / ( πm (sin d / Dirichle condiions are needed o ensure he convergence o he various inegrals, and he legiimacy o inerchanging he orders o inegraion and summaion which has been used in wriing down (] rec/ls 9

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed. Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Newton's second law in action

Newton's second law in action Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

More information

6.003: Signals and Systems

6.003: Signals and Systems 6.003: Signals and Sysems Fourier Represenaions Ocober 27, 20 2 Fourier Represenaions Fourier series represen signals in erms of sinusoids. leads o a new represenaion for sysems as filers. 3 Fourier Series

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations. Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

More information

Imagine a Source (S) of sound waves that emits waves having frequency f and therefore

Imagine a Source (S) of sound waves that emits waves having frequency f and therefore heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Intro to Fourier Series

Intro to Fourier Series Inro o Fourier Series Vecor decomposiion Even and Odd funcions Fourier Series definiion and examples Copyrigh 27 by M.H. Perro All righs reserved. M.H. Perro 27 Inro o Fourier Series, Slide 1 Review of

More information

5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance. 5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()

More information

Chapter 15: Superposition and Interference of Waves

Chapter 15: Superposition and Interference of Waves Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when

More information

Chapter 8 Copyright Henning Umland All Rights Reserved

Chapter 8 Copyright Henning Umland All Rights Reserved Chaper 8 Copyrigh 1997-2004 Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon

More information

POWER SUMS, BERNOULLI NUMBERS, AND RIEMANN S. 1. Power sums

POWER SUMS, BERNOULLI NUMBERS, AND RIEMANN S. 1. Power sums POWER SUMS, BERNOULLI NUMBERS, AND RIEMANN S ζ-function.. Power sus We begin wih a definiion of power sus, S (n. This quaniy is defined for posiive inegers > 0 and n > as he su of -h powers of he firs

More information

Chapter 2: Principles of steady-state converter analysis

Chapter 2: Principles of steady-state converter analysis Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

More information

Fourier Series Approximation of a Square Wave

Fourier Series Approximation of a Square Wave OpenSax-CNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier

More information

Differential Equations and Linear Superposition

Differential Equations and Linear Superposition Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

2. Waves in Elastic Media, Mechanical Waves

2. Waves in Elastic Media, Mechanical Waves 2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

Lectures # 5 and 6: The Prime Number Theorem.

Lectures # 5 and 6: The Prime Number Theorem. Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζ-funcion o skech an argumen which would give an acual formula for π( and sugges

More information

Section 7.1 Angles and Their Measure

Section 7.1 Angles and Their Measure Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

More information

Sums of generalized harmonic series and volumes

Sums of generalized harmonic series and volumes Sums o generalized harmonic series and umes by Fris eukers, Eugenio Calabi and Johan A.C. Kolk Mahemaisch Insiuu, Rijksuniversiei Urech PO ox 8, 358 TA Urech, The Neherlands Deparmen o Mahemaics, Universiy

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

1 The basic circulation problem

1 The basic circulation problem 2WO08: Graphs and Algorihms Lecure 4 Dae: 26/2/2012 Insrucor: Nikhil Bansal The Circulaion Problem Scribe: Tom Slenders 1 The basic circulaion problem We will consider he max-flow problem again, bu his

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 2013 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

More information

Fourier Series and Fourier Transform

Fourier Series and Fourier Transform Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved. 6.082 Spring 2007

More information

Acceleration Lab Teacher s Guide

Acceleration Lab Teacher s Guide Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

More information

9. Capacitor and Resistor Circuits

9. Capacitor and Resistor Circuits ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

More information

Capacitors and inductors

Capacitors and inductors Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear

More information

RC Circuit and Time Constant

RC Circuit and Time Constant ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

4.2 Trigonometric Functions; The Unit Circle

4.2 Trigonometric Functions; The Unit Circle 4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

A Mathematical Description of MOSFET Behavior

A Mathematical Description of MOSFET Behavior 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

More information

Math 201 Lecture 12: Cauchy-Euler Equations

Math 201 Lecture 12: Cauchy-Euler Equations Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

Modulation and Filtering

Modulation and Filtering Modulaion and Filering Wireless communicaion applicaion Impulse uncion deiniion and properies Fourier Transorm o Impulse, Sine, Cosine Picure analysis using Fourier Transorms Copyrigh 27 by M.H. Perro

More information

Section 5.1 The Unit Circle

Section 5.1 The Unit Circle Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P

More information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convolution. Recommended Problems. x2[n] 1 2[n] 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

More information

4 Fourier series. y(t) = h(τ)x(t τ)dτ = h(τ)e jω(t τ) dτ = h(τ)e jωτ e jωt dτ. = h(τ)e jωτ dτ e jωt = H(ω)e jωt.

4 Fourier series. y(t) = h(τ)x(t τ)dτ = h(τ)e jω(t τ) dτ = h(τ)e jωτ e jωt dτ. = h(τ)e jωτ dτ e jωt = H(ω)e jωt. 4 Fourier series Any LI sysem is compleely deermined by is impulse response h(). his is he oupu of he sysem when he inpu is a Dirac dela funcion a he origin. In linear sysems heory we are usually more

More information

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 0-7-380-7 Ifeachor

More information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

More information

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,

1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t, Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..

More information

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE. Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

More information

1. The graph shows the variation with time t of the velocity v of an object.

1. The graph shows the variation with time t of the velocity v of an object. 1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Rotational Inertia of a Point Mass

Rotational Inertia of a Point Mass Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

More information

Renewal processes and Poisson process

Renewal processes and Poisson process CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

t t t Numerically, this is an extension of the basic definition of the average for a discrete

t t t Numerically, this is an extension of the basic definition of the average for a discrete Average and alues of a Periodic Waveform: (Nofziger, 8) Begin by defining he average value of any ime-varying funcion over a ime inerval as he inegral of he funcion over his ime inerval, divided by : f

More information

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram. Flywheel. A flywheel is an inertial energy-storage device. It absorbs mechanical

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram. Flywheel. A flywheel is an inertial energy-storage device. It absorbs mechanical Flywheel A lywheel is an inerial energy-sorage device. I absorbs mechanical energy and serves as a reservoir, soring energy during he period when he supply o energy is more han he requiremen and releases

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

Module 4. Single-phase AC Circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC Circuits. Version 2 EE IIT, Kharagpur Module Single-phase AC Circuis Version EE, Kharagpur Lesson Generaion of Sinusoidal Volage Wavefor (AC) and Soe Fundaenal Conceps Version EE, Kharagpur n his lesson, firsly, how a sinusoidal wavefor (ac)

More information

A NOTE ON UNIT SYSTEMS

A NOTE ON UNIT SYSTEMS Tom Aage Jelmer NTNU eparmen of Peroleum Engineering and Applied Geophysics Inroducory remarks A NOTE ON UNIT SYSTEMS So far, all equaions have been expressed in a consisen uni sysem. The SI uni sysem

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

Making Use of Gate Charge Information in MOSFET and IGBT Data Sheets

Making Use of Gate Charge Information in MOSFET and IGBT Data Sheets Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

More information

4. The Poisson Distribution

4. The Poisson Distribution Virual Laboraories > 13. The Poisson Process > 1 2 3 4 5 6 7 4. The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy

More information

Discussion Examples Chapter 10: Rotational Kinematics and Energy

Discussion Examples Chapter 10: Rotational Kinematics and Energy Discussion Examples Chaper : Roaional Kinemaics and Energy 9. The Crab Nebula One o he mos sudied objecs in he nigh sky is he Crab nebula, he remains o a supernova explosion observed by he Chinese in 54.

More information

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613. Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

Lecture III: Finish Discounted Value Formulation

Lecture III: Finish Discounted Value Formulation Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

More information

LAB 6: SIMPLE HARMONIC MOTION

LAB 6: SIMPLE HARMONIC MOTION 1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:

More information

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009 ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

More information

Week #9 - The Integral Section 5.1

Week #9 - The Integral Section 5.1 Week #9 - The Inegral Secion 5.1 From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,

More information

Interference, Diffraction and Polarization

Interference, Diffraction and Polarization L.1 - Simple nerference Chaper L nerference, Diffracion and Polarizaion A sinusoidal wave raveling in one dimension has he form: Blinn College - Physics 2426 - Terry Honan A coshk x w L where in he case

More information

Fourier Series & The Fourier Transform

Fourier Series & The Fourier Transform Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The

More information

Permutations and Combinations

Permutations and Combinations Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

More information

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

The Torsion of Thin, Open Sections

The Torsion of Thin, Open Sections EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such

More information

Do not worry about your difficulties in mathematics, I assure you that mine are greater.

Do not worry about your difficulties in mathematics, I assure you that mine are greater. C H A P E R HE FOURIER SERIES 6 Do no worry abou your difficulies in mahemaics, I assure you ha mine are greaer. Alber Einsein Hisorical Profiles Jean Bapise Joseph Fourier (768 83), a French mahemaician,

More information

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

Application of kinematic equation:

Application of kinematic equation: HELP: See me (office hours). There will be a HW help session on Monda nigh from 7-8 in Nicholson 109. Tuoring a #10 of Nicholson Hall. Applicaion of kinemaic equaion: a = cons. v= v0 + a = + v + 0 0 a

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

More information

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 Life tables, force of mortality and standard life insurance products Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

More information

Two Compartment Body Model and V d Terms by Jeff Stark

Two Compartment Body Model and V d Terms by Jeff Stark Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

More information

11. Properties of alternating currents of LCR-electric circuits

11. Properties of alternating currents of LCR-electric circuits WS. Properies of alernaing currens of L-elecric circuis. Inroducion So-called passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science

More information

6.003 Homework #4 Solutions

6.003 Homework #4 Solutions 6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d

More information

AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 2007 Scoring Guidelines AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

More information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On the degrees of irreducible factors of higher order Bernoulli polynomials ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on

More information

( ) in the following way. ( ) < 2

( ) in the following way. ( ) < 2 Sraigh Line Moion - Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and man-made. Wrie down several of hem. Horizonal cars waer

More information