Chapter 6. Laplace Transforms
|
|
- Marjorie Lang
- 5 years ago
- Views:
Transcription
1 6- Chaper 6. Laplace Tranform 6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6.5 Convoluion. Inegral Equaion 6.6 Differeniaion and Inegraion of Tranform 6.7 Syem of ODE
2 6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6- Dirac dela funcion or uni impule funcion i defined a if δ a = = a a+ε a ε = oherwie ( a) d δ = The dela funcion can be obained by aking he limi of f k δ( a) = lim f k ( a) k Sifing propery of dela funcion a+ε a ε δ( ) = g a d g a The Laplace ranform of dela funcion. Sar from f k fk ( a) = u( a) u ( a k) k + { } Take he limi k and apply l Hopial rule o he quoien. k k a e a e lim e e lim k k k L δ a = e a
3 Example Ma-Spring yem under a quare wave Inpu i of he form of a recangular funcion 6-3 The ubidiary equaion Ue he parial fracion expanion : The invere ranform : Uing -hifing ( ) ( ) ( ) ( ) y = e + e u ( ) e e u + Example Hammer blow repone of a ma-pring yem The inpu i given by a dela funcion Solving algebraically The oluion ( ) ( ) y = e u e u
4 Example 3 Four-Terminal RLC-Nework 4 Find he oupu volage if R= Ω, L= H, C = F. The inpu i a dela funcion and curren and charge are zero a =. 6-4 The volage drop on R, L, C hould be equal o he inpu. Uing i = q ' The ubidiary equaion Uing -hifing and The oluion More on Parial Fracion F The oluion of a ubidiary equaion i of he form Y = G Parial fracion repreenaion may be needed. () Unrepeaed facor (-a) in G() Parial fracion hould be () Repeaed facor Repeaed facor 3 a in G() Parial fracion a in G() Parial fracion α +β (3) Unrepeaed complex facor Parial fracion A + A ( a) B ( a) ( a) A B C ( a) ( a) ( a) A + B α +β
5 Example 4 Unrepeaed Complex Facor. A damped ma-pring yem under a inuoidal force. r( ) = in for < <π y" + y' + y = r( ), = for >π, y y =, ' = The ubidiary equaion The oluion (6) The parial fracion of he fir erm Muliplying he common denominaor Term of like power of hould be equal on he righ and lef ide A=-, B=-, M=, N=6 Therefore he fir erm become The invere ranform (8) The invere of he econd erm of (6) i obained from (8) uing -hifing { } u π () Rewrie he hird erm of (6) 3 ( + ) The invere uing -hifing e co 4in (7) The final oluion For < <π y()= Eq. (8) + Eq. (7) For >π y()= Eq. (8) + Eq. (7) + Eq. ()
6 6.5 Convoluion. Inegral Equaion 6-6 The convoluion of wo funcion f and g i defined a ( * ) f g f τ g τ dτ : Noe he inegraion inerval Theorem Convoluion heorem If F and G are Laplace ranform of f and g, repecively, he muliplicaion FG i he Laplace ranform of he convoluion (f*g) Proof: Se p= τ, hen Calculae he muliplicaion G can be inide of F For fixed τ, inegrae from τ o. becaue and τ are independen. ( The inegraion over blue region ) The inegraion can be changed a e f g d ( * ) Some properie of convoluion
7 Example Convoluion Le H( ) =. Find h(). a 6-7 Rearrange : H( ) = a F() G() a Invere ranform : f( ) e g( ) =, = aτ a Uing convoluion heorem : h = f( ) * g e dτ ( e ) a Example Convoluion Le H( ) = +ω. Find h(). Rearrange : H( ) Invere of ( +ω ) : inω ω = +ω +ω ( +ω ) ( ) ( ) in ω in in * ω in in co ω Uing convoluion heorem : ω in xin y = / [ co( x + y) + co( x y)] Example 3 Unuual Properie of Convoluion f * f in general h = ωτ ω τ dτ ω + ω ω ω ω ( f * f) may no hold Applicaion o Nonhomogeneou Linear ODE Nonhomogeneou linear ODE in andard form y" + ay' + by = r : a and b, conan The oluion = ( + ) ( ) + ' + : Q( ) = Y a y y Q R Q The invere of he fir righ erm can be eaily obained. The invere of he econd erm, auming y y y = q τ rτ dτ = ' =, ranfer funcion + a + b The oupu i given by he convoluion of he impule repone q() and he driving force r(). Example 5 Ma-pring yem
8 Solve y" + 3 y' + y = r, for r = < < = oherwie y y = ' = 6-8 The ranfer funcion I invere Since y y = ' =, he oluion i given by he convoluion of q and r. ( ) ( ) ( τ) ( τ) ( τ) ( τ) y = q τ rτ dτ q τ uτ uτ dτ e e dτ e e For < : y() = r()= only for << For << : The upper limi i, For > : The upper limi i, Noe he change in he lower limi. hould be le han. τ= ( τ) ( τ) ( ) ( ) y = e e e e + τ= τ= y = e e e e e e τ= τ= τ= ( τ) ( τ) ( ) ( ) ( ) ( ) Inegral Equaion Convoluion can be ued o olve cerain inegral equaion Example 6 A volerra Inegral Equaion of he Second Kind Solve Convoluion of y( ) and in( ) Uing he convoluion heorem we obain he ubidiary equaion + Y( ) Y( ) = Y ( ) Y ( ) = The anwer i
9 6.6 Differeniaion and Inegraion of Tranform. 6-9 Differeniaion of Tranform If F() i he ranform of f(), hen i derivaive i df ( ) F ( ) = f ( ) e d F '( ) = = f ( ) e d d Conequenly L f( ) = F' ( ) and L F' ( ) = f ( ) Example Differeniaion of Tranform The able can be proved uing differeniaion of F(). The econd one d β β L [ inβ ] = d +β +β Inegraion of Tranform If f() ha a ranform and f( ) lim / exi, + f( ) = L F ( ) d and L F d = f Proof: From he definiion f( ) F ( ) d = e f ( ) d d f ( ) e d d L Revere he order of inegraion. = e /
10 Example Differeniaion and Inegraion of Tranform 6- Find he invere ranform of F() = I derivaive Take he invere ranform L F ' L coω = f +ω f( ) coω = Uing inegraion of ranform Le Then = ( ) ' F F F d G d Take he invere ranform of boh ide g( ) coω f( ) = f( ) = Special Linear ODE wih Variable Coefficien Ue differeniaion of ranform o olve ODE. Le L [ y] = Y [ y' ] = Y y( ) L. Uing differeniaion of ranform d dy L [ y' ] = Y y( ) Y d d Similarly, uing L [ y" ] = Y y( ) y' L d d dy d [ y" ] = Y y( ) y' Y + y( )
11 Example 3 Laguerre Equaion Laguerre ODE i y" + y' + ny = n=,,, 6- The ubidiary equaion Separaing variable, uing parial fracion dy n + n n + = d d Y ( ) lny = nln n+ ln ln n + n Y = ( ) n n + The invere ranform i given by Rodrigue formula l = L Y n [ ] n=,, Prove Rodrigue formula Uing -hifing Uing he n-h derivaive of f, Afer anoher -hifing
12 6.7 Syem of ODE 6- The Laplace ranform can be ued o olve yem of ODE. Conider a fir-order linear yem wih conan coefficien The ubidiary equaion Rearrange a Y+ a Y = y G ( ) a Y+ ( a Y ) = y( ) G( ) Solve hi yem algebraically for Y ( ) and Y ( ) and ake he invere ranform for y ( ) and y ( ) Example Elecrical Nework i and i. Find he curren v( ) = vol only for.5 and i =' i = From Kirchhoff volage law in he lower and he upper circui, Rearrange i = i = The ubidiary equaion uing Solve algebraically for I and I ( + ) I = e e 7 7 ( / ) ( / ) I = e + e 7 7 ( / ) ( / )
13 The invere ranform of he quare bracke erm uing -hifing. 5 5 / 65 7 / e e / 5 7 / e + e Uing -hifing 5 5 / 65 7 / 5 5 (.5)/ 65 7(.5)/ i ( ) = e e e e u (.5) / 5 7 / 5 5 (.5)/ 5 7(.5)/ i ( ) = e + e e + e u (.5) Noe ha he oluion for i differen from ha for due o he uni ep funcion. Example 3 Two mae on Spring Ignoring he ma of he pring and he damping Newon econd law(ma X acceleraion) Hooke law (reoring force) Iniial condiion y = y = y ' = 3 k, y ' = 3k The ubidiary equaion The algebraic oluion uing Cramer rule The final oluion
Chapter 6. Laplace Transforms
Chaper 6. Laplace Tranform Kreyzig by YHLee;45; 6- An ODE i reduced o an algebraic problem by operaional calculu. The equaion i olved by algebraic manipulaion. The reul i ranformed back for he oluion of
More information6.8 Laplace Transform: General Formulas
48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More informationCHAPTER. Forced Equations and Systems { } ( ) ( ) 8.1 The Laplace Transform and Its Inverse. Transforms from the Definition.
CHAPTER 8 Forced Equaion and Syem 8 The aplace Tranform and I Invere Tranform from he Definiion 5 5 = b b {} 5 = 5e d = lim5 e = ( ) b {} = e d = lim e + e d b = (inegraion by par) = = = = b b ( ) ( )
More informationLaplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)
Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful
More information18.03SC Unit 3 Practice Exam and Solutions
Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care
More informationCONTROL SYSTEMS. Chapter 3 Mathematical Modelling of Physical Systems-Laplace Transforms. Prof.Dr. Fatih Mehmet Botsalı
CONTROL SYSTEMS Chaper Mahemaical Modelling of Phyical Syem-Laplace Tranform Prof.Dr. Faih Mehme Boalı Definiion Tranform -- a mahemaical converion from one way of hinking o anoher o make a problem eaier
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
More information13.1 Circuit Elements in the s Domain Circuit Analysis in the s Domain The Transfer Function and Natural Response 13.
Chaper 3 The Laplace Tranform in Circui Analyi 3. Circui Elemen in he Domain 3.-3 Circui Analyi in he Domain 3.4-5 The Tranfer Funcion and Naural Repone 3.6 The Tranfer Funcion and he Convoluion Inegral
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More informationLaplace Transforms. Examples. Is this equation differential? y 2 2y + 1 = 0, y 2 2y + 1 = 0, (y ) 2 2y + 1 = cos x,
Laplace Transforms Definiion. An ordinary differenial equaion is an equaion ha conains one or several derivaives of an unknown funcion which we call y and which we wan o deermine from he equaion. The equaion
More informationChapter 9 - The Laplace Transform
Chaper 9 - The Laplace Tranform Selece Soluion. Skech he pole-zero plo an region of convergence (if i exi) for hee ignal. ω [] () 8 (a) x e u = 8 ROC σ ( ) 3 (b) x e co π u ω [] ( ) () (c) x e u e u ROC
More informationCHAPTER 3 SIGNALS & SYSTEMS. z -transform in the z -plane will be (A) 1 (B) 1 (D) (C) . The unilateral Laplace transform of tf() (A) s (B) + + (D) (C)
CHAPER SIGNALS & SYSEMS YEAR ONE MARK n n MCQ. If xn [ ] (/) (/) un [ ], hen he region of convergence (ROC) of i z ranform in he z plane will be (A) < z < (B) < z < (C) < z < (D) < z MCQ. he unilaeral
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationEE Control Systems LECTURE 2
Copyrigh F.L. Lewi 999 All righ reerved EE 434 - Conrol Syem LECTURE REVIEW OF LAPLACE TRANSFORM LAPLACE TRANSFORM The Laplace ranform i very ueful in analyi and deign for yem ha are linear and ime-invarian
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationSEC. 6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem (t-shifting) 217. Second Shifting Theorem (t-shifting)
SEC. 6.3 Uni Sep Funcion (Heaviside Funcion). Second Shifing Theorem (-Shifing) 7. PROJECT. Furher Resuls by Differeniaion. Proceeding as in Example, obain (a) and from his and Example : (b) formula, (c),
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationEE202 Circuit Theory II
EE202 Circui Theory II 2017-2018, Spring Dr. Yılmaz KALKAN I. Inroducion & eview of Fir Order Circui (Chaper 7 of Nilon - 3 Hr. Inroducion, C and L Circui, Naural and Sep epone of Serie and Parallel L/C
More informationChapter 7: Inverse-Response Systems
Chaper 7: Invere-Repone Syem Normal Syem Invere-Repone Syem Baic Sar ou in he wrong direcion End up in he original eady-ae gain value Two or more yem wih differen magniude and cale in parallel Main yem
More informationENGI 9420 Engineering Analysis Assignment 2 Solutions
ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion
More informationMore on ODEs by Laplace Transforms October 30, 2017
More on OE b Laplace Tranfor Ocober, 7 More on Ordinar ifferenial Equaion wih Laplace Tranfor Larr areo Mechanical Engineering 5 Seinar in Engineering nali Ocober, 7 Ouline Review la cla efiniion of Laplace
More informationInstrumentation & Process Control
Chemical Engineering (GTE & PSU) Poal Correpondence GTE & Public Secor Inrumenaion & Proce Conrol To Buy Poal Correpondence Package call a -999657855 Poal Coure ( GTE & PSU) 5 ENGINEERS INSTITUTE OF INDI.
More informationCONTROL SYSTEMS. Chapter 10 : State Space Response
CONTROL SYSTEMS Chaper : Sae Space Repone GATE Objecive & Numerical Type Soluion Queion 5 [GATE EE 99 IIT-Bombay : Mark] Conider a econd order yem whoe ae pace repreenaion i of he form A Bu. If () (),
More informationChapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More information1 CHAPTER 14 LAPLACE TRANSFORMS
CHAPTER 4 LAPLACE TRANSFORMS 4 nroducion f x) i a funcion of x, where x lie in he range o, hen he funcion p), defined by p) px e x) dx, 4 i called he Laplace ranform of x) However, in hi chaper, where
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are Coninuous-Time Signals??? Reading Assignmen: Secion. of Kamen and Heck /22 Course Flow Diagram The arrows here show concepual flow beween ideas.
More informationNODIA AND COMPANY. GATE SOLVED PAPER Electrical Engineering SIGNALS & SYSTEMS. Copyright By NODIA & COMPANY
No par of hi publicaion may be reproduced or diribued in any form or any mean, elecronic, mechanical, phoocopying, or oherie ihou he prior permiion of he auhor. GAE SOLVED PAPER Elecrical Engineering SIGNALS
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationExponential Sawtooth
ECPE 36 HOMEWORK 3: PROPERTIES OF THE FOURIER TRANSFORM SOLUTION. Exponenial Sawooh: The eaie way o do hi problem i o look a he Fourier ranform of a ingle exponenial funcion, () = exp( )u(). From he able
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationt )? How would you have tried to solve this problem in Chapter 3?
Exercie 9) Ue Laplace ranform o wrie down he oluion o 2 x x = F in x = x x = v. wha phenomena do oluion o hi DE illurae (even hough we're forcing wih in co )? How would you have ried o olve hi problem
More informationu(t) Figure 1. Open loop control system
Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference
More informationSerial : 4LS1_A_EC_Signal & Systems_230918
Serial : LS_A_EC_Signal & Syem_8 CLASS TEST (GATE) Delhi oida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubanewar Kolkaa Pana Web: E-mail: info@madeeay.in Ph: -56 CLASS TEST 8- ELECTROICS EGIEERIG Subjec
More informationThe complex Fourier series has an important limiting form when the period approaches infinity, i.e., T 0. 0 since it is proportional to 1/L, but
Fourier Transforms The complex Fourier series has an imporan limiing form when he period approaches infiniy, i.e., T or L. Suppose ha in his limi () k = nπ L remains large (ranging from o ) and (2) c n
More informationLectures 29 and 30 BIQUADRATICS AND STATE SPACE OP AMP REALIZATIONS. I. Introduction
EE-202/445, 3/18/18 9-1 R. A. DeCarlo Lecures 29 and 30 BIQUADRATICS AND STATE SPACE OP AMP REALIZATIONS I. Inroducion 1. The biquadraic ransfer funcion has boh a 2nd order numeraor and a 2nd order denominaor:
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 30 Signal & Syem Prof. ark Fowler oe Se #34 C-T Tranfer Funcion and Frequency Repone /4 Finding he Tranfer Funcion from Differenial Eq. Recall: we found a DT yem Tranfer Funcion Hz y aking he ZT of
More informationMon Apr 2: Laplace transform and initial value problems like we studied in Chapter 5
Mah 225-4 Week 2 April 2-6 coninue.-.3; alo cover par of.4-.5, EP 7.6 Mon Apr 2:.-.3 Laplace ranform and iniial value problem like we udied in Chaper 5 Announcemen: Warm-up Exercie: Recall, The Laplace
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2008
[E5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 008 EEE/ISE PART II MEng BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: :00 hours There are FOUR quesions
More informationLaplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More informationPiecewise-Defined Functions and Periodic Functions
28 Piecewie-Defined Funcion and Periodic Funcion A he ar of our udy of he Laplace ranform, i wa claimed ha he Laplace ranform i paricularly ueful when dealing wih nonhomogeneou equaion in which he forcing
More informations-domain Circuit Analysis
Domain ircui Analyi Operae direcly in he domain wih capacior, inducor and reior Key feaure lineariy i preerved c decribed by ODE and heir I Order equal number of plu number of Elemenbyelemen and ource
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationCHAPTER 7. Definition and Properties. of Laplace Transforms
SERIES OF CLSS NOTES FOR 5-6 TO INTRODUCE LINER ND NONLINER PROBLEMS TO ENGINEERS, SCIENTISTS, ND PPLIED MTHEMTICINS DE CLSS NOTES COLLECTION OF HNDOUTS ON SCLR LINER ORDINRY DIFFERENTIL EQUTIONS (ODE")
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationDelhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Serial : 0. ND_NW_EE_Signal & Sysems_4068 Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkaa Pana Web: E-mail: info@madeeasy.in Ph: 0-4546 CLASS TEST 08-9 ELECTRICAL ENGINEERING
More informationSignals and Systems Linear Time-Invariant (LTI) Systems
Signals and Sysems Linear Time-Invarian (LTI) Sysems Chang-Su Kim Discree-Time LTI Sysems Represening Signals in Terms of Impulses Sifing propery 0 x[ n] x[ k] [ n k] k x[ 2] [ n 2] x[ 1] [ n1] x[0] [
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationln 2 1 ln y x c y C x
Lecure 14 Appendi B: Some sample problems from Boas Here are some soluions o he sample problems assigned for Chaper 8 8: 6 Soluion: We wan o find he soluion o he following firs order equaion using separaion
More informationEEEB113 CIRCUIT ANALYSIS I
9/14/29 1 EEEB113 CICUIT ANALYSIS I Chaper 7 Firs-Order Circuis Maerials from Fundamenals of Elecric Circuis 4e, Alexander Sadiku, McGraw-Hill Companies, Inc. 2 Firs-Order Circuis -Chaper 7 7.2 The Source-Free
More informationEE202 Circuit Theory II , Spring. Dr. Yılmaz KALKAN & Dr. Atilla DÖNÜK
EE202 Circui Theory II 2018 2019, Spring Dr. Yılmaz KALKAN & Dr. Ailla DÖNÜK 1. Basic Conceps (Chaper 1 of Nilsson - 3 Hrs.) Inroducion, Curren and Volage, Power and Energy 2. Basic Laws (Chaper 2&3 of
More informationCHAPTER 6: FIRST-ORDER CIRCUITS
EEE5: CI CUI T THEOY CHAPTE 6: FIST-ODE CICUITS 6. Inroducion This chaper considers L and C circuis. Applying he Kirshoff s law o C and L circuis produces differenial equaions. The differenial equaions
More informationwhere the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).
Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness
More informationOn the Exponential Operator Functions on Time Scales
dvance in Dynamical Syem pplicaion ISSN 973-5321, Volume 7, Number 1, pp. 57 8 (212) hp://campu.m.edu/ada On he Exponenial Operaor Funcion on Time Scale laa E. Hamza Cairo Univeriy Deparmen of Mahemaic
More informationCHAPTER 2: Mathematics for Microeconomics
CHAPTER : Mahemaics for Microeconomics The problems in his chaper are primarily mahemaical. They are inended o give sudens some pracice wih he conceps inroduced in Chaper, bu he problems in hemselves offer
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationA Theoretical Model of a Voltage Controlled Oscillator
A Theoreical Model of a Volage Conrolled Ocillaor Yenming Chen Advior: Dr. Rober Scholz Communicaion Science Iniue Univeriy of Souhern California UWB Workhop, April 11-1, 6 Inroducion Moivaion The volage
More informationδ (τ )dτ denotes the unit step function, and
ECE-202 Homework Problems (Se 1) Spring 18 TO THE STUDENT: ALWAYS CHECK THE ERRATA on he web. ANCIENT ASIAN/AFRICAN/NATIVE AMERICAN/SOUTH AMERICAN ETC. PROVERB: If you give someone a fish, you give hem
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationLecture 13 RC/RL Circuits, Time Dependent Op Amp Circuits
Lecure 13 RC/RL Circuis, Time Dependen Op Amp Circuis RL Circuis The seps involved in solving simple circuis conaining dc sources, resisances, and one energy-sorage elemen (inducance or capaciance) are:
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationChapter 4 The Fourier Series and Fourier Transform
Represenaion of Signals in Terms of Frequency Componens Chaper 4 The Fourier Series and Fourier Transform Consider he CT signal defined by x () = Acos( ω + θ ), = The frequencies `presen in he signal are
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationEE 313 Linear Signals & Systems (Fall 2018) Solution Set for Homework #8 on Continuous-Time Signals & Systems
EE 33 Linear Signals & Sysems (Fall 08) Soluion Se for Homework #8 on Coninuous-Time Signals & Sysems By: Mr. Houshang Salimian & Prof. Brian L. Evans Here are several useful properies of he Dirac dela
More informationDirect Current Circuits. February 19, 2014 Physics for Scientists & Engineers 2, Chapter 26 1
Direc Curren Circuis February 19, 2014 Physics for Scieniss & Engineers 2, Chaper 26 1 Ammeers and Volmeers! A device used o measure curren is called an ammeer! A device used o measure poenial difference
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More information, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max
ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen
More informationR =, C = 1, and f ( t ) = 1 for 1 second from t = 0 to t = 1. The initial charge on the capacitor is q (0) = 0. We have already solved this problem.
Theoreical Physics Prof. Ruiz, UNC Asheville, docorphys on YouTube Chaper U Noes. Green's Funcions R, C 1, and f ( ) 1 for 1 second from o 1. The iniial charge on he capacior is q (). We have already solved
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More informationLAPLACE TRANSFORMS. 1. Basic transforms
LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se # Wha are Coninuous-Time Signals??? /6 Coninuous-Time Signal Coninuous Time (C-T) Signal: A C-T signal is defined on he coninuum of ime values. Tha is:
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
More information6 December 2013 H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1
Lecure Noe Fundamenal of Conrol Syem Inrucor: Aoc. Prof. Dr. Huynh Thai Hoang Deparmen of Auomaic Conrol Faculy of Elecrical & Elecronic Engineering Ho Chi Minh Ciy Univeriy of Technology Email: hhoang@hcmu.edu.vn
More informationLinear Time-invariant systems, Convolution, and Cross-correlation
Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 39
ECE 6341 Spring 2016 Prof. David R. Jackon ECE Dep. Noe 39 1 Finie Source J ( y, ) For a phaed curren hee: p p j( k kyy) J y = J e + (, ) 0 The angenial elecric field ha i produced i: ( p ˆ ˆ) j( k + k
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More information6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson
6.32 Feedback Syem Phae-locked loop are a foundaional building block for analog circui deign, paricularly for communicaion circui. They provide a good example yem for hi cla becaue hey are an excellen
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More information8.1. a) For step response, M input is u ( t) Taking inverse Laplace transform. as α 0. Ideal response, K c. = Kc Mτ D + For ramp response, 8-1
8. a For ep repone, inpu i u, U Y a U α α Y a α α Taking invere Laplae ranform a α e e / α / α A α 0 a δ 0 e / α a δ deal repone, α d Y i Gi U i δ Hene a α 0 a i For ramp repone, inpu i u, U Soluion anual
More informationcan be viewed as a generalized product, and one for which the product of f and g. That is, does
Boyce/DiPrim 9 h e, Ch 6.6: The Convoluion Inegrl Elemenry Differenil Equion n Bounry Vlue Problem, 9 h eiion, by Willim E. Boyce n Richr C. DiPrim, 9 by John Wiley & Son, Inc. Someime i i poible o wrie
More informationAbout the HELM Project HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-year curriculum development project
Abou he HELM Projec HELM (Helping Engineers Learn Mahemaics) maerials were he oucome of a hree-year curriculum developmen projec underaken by a consorium of five English universiies led by Loughborough
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More informationChapter One Fourier Series and Fourier Transform
Chaper One I. Fourier Series Represenaion of Periodic Signals -Trigonomeric Fourier Series: The rigonomeric Fourier series represenaion of a periodic signal x() x( + T0 ) wih fundamenal period T0 is given
More information2 int T. is the Fourier transform of f(t) which is the inverse Fourier transform of f. i t e
PHYS67 Class 3 ourier Transforms In he limi T, he ourier series becomes an inegral ( nt f in T ce f n f f e d, has been replaced by ) where i f e d is he ourier ransform of f() which is he inverse ourier
More informationContinuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mar Fowler Noe Se #1 C-T Signals: Circuis wih Periodic Sources 1/1 Solving Circuis wih Periodic Sources FS maes i easy o find he response of an RLC circui o a periodic source!
More informationLaplace Transform and its Relation to Fourier Transform
Chaper 6 Laplace Transform and is Relaion o Fourier Transform (A Brief Summary) Gis of he Maer 2 Domains of Represenaion Represenaion of signals and sysems Time Domain Coninuous Discree Time Time () [n]
More information