e st f (t) dt = e st tf(t) dt = L {t f(t)} s
|
|
- Alaina Quinn
- 6 years ago
- Views:
Transcription
1 Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic function? Multiplying a function by t n d ds F(s) = d e st f (t) dt = ds that is e st f (t) dt = e st tf(t) dt = L {t f(t)} s Similarly L {t f(t)} = d ds L { f (t)} L t 2 f (t) = L {t.t f(t)} = d ds L {t f(t)} = d ds dds L { f (t)} = d2 ds2l { f (t)}
2 Theorem: Derivatives of transforms If F(s) = L { f (t)} and n =, 2, 3,... then Example : L {t sin kt} L t n f (t) = ( ) n dn dsnf(s) () With f (t) = sin kt, F(s) = k/(s 2 + k 2 ), and n =, the theorem above gives L {t sin kt} = d ds L {sin kt} = d k 2ks ds s 2 + k 2 = (s 2 + k 2 ) 2 Evaluate L t 2 sin kt and L t 3 sin kt.
3 Example 2: x + 6x = cos 4t, x() =, x () = The Laplace transform of the DE gives (s 2 + 6)X(s) = + X(s) = s s s s (s 2 + 6) 2 In the example we have got L 2ks (s 2 + k 2 ) 2 = t sin kt and so with the identification k = 4, we obtain x(t) = 4 4 L s s + 6 L (s 2 + 6) 2 = 4 sin 4t + t sin 4t 8
4 Transforms of Integrals Convolution If functions f and g are piecewise continuous on [, ) then a special product, f g, defined by the integral t f g = f (τ)g(t τ)dτ (2) is called the convolution of f and g. The convolution f g is a function of t. Example: e t sin t = t e τ sin(t τ) dτ = 2 sin t cos t + e t
5 The convolution of two functions is commutative: t t f g = f (τ)g(t τ)dτ = f (t τ)g(τ)dτ = g f (3) The Laplace transform of the convolution f g is the product of Laplace transforms of f and g, thus we can find the Laplace transform of the convolution without performing the convolution integral. Convolution theorem If f (t) and g(t) are piecewise continuous on [, ) and of exponential order, then L { f g} = L { f (t)} L {g(t)} = F(s)G(s). (4) The Proof: see D.G. Zill et al., Advanced Engineering Mathematics, 4th Ed., p.222.
6 Example 3 L e t sin t t = L e τ sin(t τ) dτ = L e t L {sin t} = s. s 2 + = (s )(s 2 + ) Inverse form of the convolution theorem L {F(s)G(s)} = f g (5)
7 Example 4: L (s 2 +k 2 ) 2 Let then and F(s) = G(s) = s 2 + k 2 f (t) = g(t) = k k L s 2 + k 2 = sin kt k L (s 2 + k 2 ) 2 = k t 2 sin kτ sin k(t τ)dτ
8 Using sin A sin B = [cos(a B) cos(a + B)] (6) 2 with A = kτ and B = k(t τ), we can carry out the integration L (s 2 + k 2 ) 2 = t k 2 sin kτ sin k(t τ)dτ = = = 2k 2 2k 2 t [cos k(2τ t) cos kt]dτ sin k(2τ t) τ cos kt 2k sin kt kt cos kt 2k 3 t
9 Transform of an integral When g(t) = and L {g(t)} = G(s) = /s, the convolution theorem implies that the Laplace transform of the integral of f is t L f (τ) dτ = F(s) s (7) The inverse form of the equation above t f (τ) dτ = L F(s) s can be used instead of partial fractions when s n is a factor of the denominator and f (t) = L {F(s)} is easy to integrate. (8)
10 Examples: L s(s 2 + ) L s 2 (s 2 + ) L s 3 (s 2 + ) = L = L = L /(s 2 + ) t s = /s(s 2 + ) s /s 2 (s 2 + ) s sin τ dτ = cos t t = ( cos τ) dτ = t sin t t = (τ sin τ) dτ = 2 t2 + cos t and so on.
11 Volterra integral equation The convolution theorem and the Laplace transform of integrals are useful for solving types of equations in which an unknown function appears under an integral sign. For example, we can solve a Volterra integral equation for f (t) f (t) = g(t) + t f (τ)h(t τ) dτ (9) The functions g(t) and h(t) are known. Notice that the integral above is of the convolution form.
12 Example 5: An integral equation Solve t f (t) = 3t 2 e t for f (t). f (τ) e t τ dτ We identify h(t τ) = e t τ, so h(t) = e t and take the Laplace transform of each term; in particular the integral transforms as the product of L { f (t)} = F(s) and L e t = /(s ): F(s) = 3. 2 s 3 s + F(s). s
13 After solving the equation for F(s) and carrying the partial fraction decomposition, we get F(s) = 6 s 3 6 s 4 + s 2 s + the inverse Laplace transform then gives f (t) = 3L 2! s 3 L 3! s 4 = 3t 2 t 3 + 2e t + L s 2L s +
14 Series circuits Recall that in a single-loop or series circuits, Kirchhoff s second law states that the sum of the voltage drops across an inductor, resistor, and capacitor is equal to the impressed voltage E(t). The voltage drops across an inductor, resistor and capacitor are, respectively, L di dt, Ri(t), and C t i(τ) dτ where i(t) is the current, and L, R, and C are constants. It follows that the current in a circuit is govern by the integrodifferential equation L di dt + Ri(t) + t i(τ) dτ = E(t) C
15 Example 6: An integrodifferential equation Determine the current i(t) in a single-loop LRC-circuit when L =. h, R = 2 Ω, C =. f, i() =, and the impressed voltage is E(t) = 2 t 2 t U(t ) Using the data above, the integrodifferential equation we are to solve is. di t + 2 i(t) + i(τ) dτ = 2 t 2 t U(t ) dt
16 Using L t i(τ) dτ = I(s)/s, where I(s) = L {i(t)}, the Laplace transform of the equation is. si(s) + 2 I(s) + I(s) s = 2 s 2 s 2e s s e s where we have used L {g(t)u(t a)} = e as L {g(t + a)} (see the section on the second translation theorem). Solving the equation above for I(s) yields I(s) = 2 s(s + ) 2 s(s + ) 2e s (s + ) 2e s / = 2 / s s + / (s + ) 2 / e s s + / s + e s + / (s + ) 2e s (s + ) 2e s
17 From the inverse form of the second translation theorem, we obtain the current i(t) = 2 [ U (t )] 2 e t e (t ) U (t ) 2te t 8(t )e (t ) U (t ) which can be rewritten as the piecewise-defined function i(t) = 2 2e t 2te t, t < 2e t + 2e (t ) 2e t 8(t )e (t ), t.
18 Transform of a periodic function Periodic function If a periodic function f has period T, T >, then f (t + T) = f (t). Theorem: Transform of a periodic function If f (t) is a piecewise continuous on [, ), of exponential order, and periodic with period T, the L { f (t)} = e st T e st f (t) dt The Proof: see D.G. Zill et al., Advanced Engineering Mathematics, 4th Ed., p.226.
19 Example 7: Transform of a periodic function Find the Laplace transform of the square wave E(t) with the period T = 2. t < 2, it can be defined by For E(t) =, t <, t < 2. and outside of the interval by f (t + 2) = f (t). From the theorem above, we have L {E(t)} = 2 e 2s e st 2 E(t) dt = e 2s e st dt + e st dt = e s e s e 2s = s ( + e s )( e s ) s = s ( + e s )
20 Example 8: A periodic impressed voltage Determine the current i(t) in a single-loop LR-series circuit L di + Ri= E(t) () dt where E(t) is the same function that we studied in the previous example. The Laplace transform of the differential equation is or LsI(s) + RI(s) = I(s) = s ( + e s ) /L s(s + R/L) + e s
21 To find the inverse Laplace transform of the last expression, we identify x = e s, s >, and use the geometric series + x = x + x2 x e s = e s + e 2s e 3s +... From the partial fraction decomposition we can rewrite I(s) as I(s) = R s s + R/L = R s e s s + e 2s s s(s + R/L) = L/R s L/R s + R/L e s + e 2s e 3s +... e 3s s +... R s + R/L e s s + R/L + e 2s s + R/L e 3s s + R/L +...
22 By applying the second translation theorem to each term of both series we get i(t) = ( U (t ) + U (t 2) U (t 3) +...) R e Rt/L e R(t )/L U (t ) + e R(t 2)/L U (t 2) e R(t 3)/L U (t 3) +... R or equivalently i(t) = e Rt/L + R R ( ) n e R(t n)/l U (t n) n=
23 To interpret the solution, let us assume for illustration that R =, L = and t < 4. We get i(t) = e t e (t ) U (t ) + e (t 2) U (t 2) e (t 3) U (t 3) or in other words i(t) = e t, t < e t + e (t ), t < 2 e t + e (t ) e (t 2), 2 t < 3 e t + e (t ) e (t 2) + e (t 3), 3 t < 4
24 The Dirac delta function Unit pulse, t < t a δ a (t t ) = 2a, t a t < t + a, t t + a a >, t >. For a small value of a, δ a (t t ) is essentially a constant function of large magnitude that is on for just a very short period of time, around t. As a the magnitude of the unit pulse grows to while the area under the unit pulse is constant and equals to : δ a (t t ) dt =
25 The Dirac delta function is defined as the limit of the unit pulse δ(t t ) = lim a δ a (t t ) () and it can be characterized by two properties: (i) (ii) δ(t t ) =, t = t, t t δ(t t ) dt =.
26 It is possible to obtain the Laplace transform of the Dirac delta function by the formal assumption that L δ(t t ) = lim a L δ a (t t ). Theoremn: Transform of the Dirac delta function For t > L δ(t t ) = e st (2) Proof: δ a (t t ) = U t (t a) U t (t 2a + a) L δ a (t t ) = e s(t a) e s(t +a) = e st e sa e sa 2a s s 2sa L δ(t t ) = lim L δ a (t t ) = e st e sa e sa lim = e st a a 2sa Now, when t =, L {δ(t)} =. This emphasizes the fact that the delta function is not a usual type of function, since we expect L { f (t)} as s.
27 Example: Two initial value problems (the delta-function kicked oscillator) Solve y + y = 4δ(t 2π) subject to (a) y() =, y () =, and (b) y() =, y () =. (a) The Laplace transform of the DE is s 2 Y(s) s + Y(s) = 4e 2πs s Y(s) = s e 2πs s 2 + Using the inverse form of the second translation theorem, we get y(t) = cos t + 4 sin(t 2π)U (t 2π) and since sin(t 2π) = sin t, the solution can be rewritten as y(t) = cos t, t < 2π cos t + 4 sin t, t 2π.
28 (b) The transform of the DE in this case is simpler and so the final result is Y(s) = 4e 2πs s 2 + y(t) = 4 sin(t 2π)U (t 2π) or y(t) =, t < 2π 4 sin t, t 2π.
29 Remarks: (i) The delta function is not a proper function but rather an example of distribution. It is best characterized by its effect on the other functions: if f is a continuous function then f (t) δ(t t ) dt = f (t ) This is known as the sifting property. (ii) Recall that the transfer function of a general nth-order DE with constant coefficients is W(s) = /P(s) where P(s) = a n s n + a n s n a. The transfer function is the Laplace transform of function w(t) called the weight function of a linear system. How does this work if the input is impulsive, i.e. it has the form of the delta function?
30 Consider the second-order linear system in which the input is the unit impulse at t = a 2 y + a y + a y = δ(t), y() =, y () =. Applying the Laplace transform and using L {δ(t)} = shows that the transform of the response y in this case is the transfer function and so Y(s) = a 2 s 2 = + a s + a P(s) = W(s) y = L P(s) = w(t) Thus, in general, the weight function y = w(t) of an nth-order linear system is the zero-state response of the system to a unit pulse, and for this reason w(t) is called the impulse response of the system.
f(t)e st dt. (4.1) Note that the integral defining the Laplace transform converges for s s 0 provided f(t) Ke s 0t for some constant K.
4 Laplace transforms 4. Definition and basic properties The Laplace transform is a useful tool for solving differential equations, in particular initial value problems. It also provides an example of integral
More informationENGIN 211, Engineering Math. Laplace Transforms
ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving
More information(f g)(t) = Example 4.5.1: Find f g the convolution of the functions f(t) = e t and g(t) = sin(t). Solution: The definition of convolution is,
.5. Convolutions and Solutions Solutions of initial value problems for linear nonhomogeneous differential equations can be decomposed in a nice way. The part of the solution coming from the initial data
More informationLecture 7: Laplace Transform and Its Applications Dr.-Ing. Sudchai Boonto
Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkut s Unniversity of Technology Thonburi Thailand Outline Motivation The Laplace Transform The Laplace Transform
More informationAdvanced Analog Building Blocks. Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc
Advanced Analog Building Blocks Prof. Dr. Peter Fischer, Dr. Wei Shen, Dr. Albert Comerma, Dr. Johannes Schemmel, etc 1 Topics 1. S domain and Laplace Transform Zeros and Poles 2. Basic and Advanced current
More informationLaplace Transform. Chapter 4
Chapter 4 Laplace Transform It s time to stop guessing solutions and find a systematic way of finding solutions to non homogeneous linear ODEs. We define the Laplace transform of a function f in the following
More informationMath 307 Lecture 19. Laplace Transforms of Discontinuous Functions. W.R. Casper. Department of Mathematics University of Washington.
Math 307 Lecture 19 Laplace Transforms of Discontinuous Functions W.R. Casper Department of Mathematics University of Washington November 26, 2014 Today! Last time: Step Functions This time: Laplace Transforms
More informationLTI Systems (Continuous & Discrete) - Basics
LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying
More informationThe Laplace transform
The Laplace transform Samy Tindel Purdue University Differential equations - MA 266 Taken from Elementary differential equations by Boyce and DiPrima Samy T. Laplace transform Differential equations 1
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationLaplace Transforms and use in Automatic Control
Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More informationSection 6.4 DEs with Discontinuous Forcing Functions
Section 6.4 DEs with Discontinuous Forcing Functions Key terms/ideas: Discontinuous forcing function in nd order linear IVPs Application of Laplace transforms Comparison to viewing the problem s solution
More informationThe Laplace Transform
C H A P T E R 6 The Laplace Transform Many practical engineering problems involve mechanical or electrical systems acted on by discontinuous or impulsive forcing terms. For such problems the methods described
More informationDifferential Equations
Differential Equations Math 341 Fall 21 MWF 2:3-3:25pm Fowler 37 c 21 Ron Buckmire http://faculty.oxy.edu/ron/math/341/1/ Worksheet 29: Wednesday December 1 TITLE Laplace Transforms and Introduction to
More informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
More informationSource-Free RC Circuit
First Order Circuits Source-Free RC Circuit Initial charge on capacitor q = Cv(0) so that voltage at time 0 is v(0). What is v(t)? Prof Carruthers (ECE @ BU) EK307 Notes Summer 2018 150 / 264 First Order
More informationMath 3313: Differential Equations Laplace transforms
Math 3313: Differential Equations Laplace transforms Thomas W. Carr Department of Mathematics Southern Methodist University Dallas, TX Outline Introduction Inverse Laplace transform Solving ODEs with Laplace
More informationComputing inverse Laplace Transforms.
Review Exam 3. Sections 4.-4.5 in Lecture Notes. 60 minutes. 7 problems. 70 grade attempts. (0 attempts per problem. No partial grading. (Exceptions allowed, ask you TA. Integration table included. Complete
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More informationMath 308 Exam II Practice Problems
Math 38 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationTaking the Laplace transform of the both sides and assuming that all initial conditions are zero,
The transfer function Let s begin with a general nth-order, linear, time-invariant differential equation, d n a n dt nc(t)... a d dt c(t) a 0c(t) d m = b m dt mr(t)... a d dt r(t) b 0r(t) () where c(t)
More informationGeneralized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks:
Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n, d n, t < δ n (t) = n, t 3 d3 d n, t > n. d t The Dirac delta generalized function
More informationLecture 6: Time-Domain Analysis of Continuous-Time Systems Dr.-Ing. Sudchai Boonto
Lecture 6: Time-Domain Analysis of Continuous-Time Systems Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkut s Unniversity of Technology Thonburi Thailand
More informationUnit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace
Unit : Modeling in the Frequency Domain Part : Engineering 81: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 1, 010 1 Pair Table Unit, Part : Unit,
More information+ + LAPLACE TRANSFORM. Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions.
COLOR LAYER red LAPLACE TRANSFORM Differentiation & Integration of Transforms; Convolution; Partial Fraction Formulas; Systems of DEs; Periodic Functions. + Differentiation of Transforms. F (s) e st f(t)
More informationLaplace Transform Part 1: Introduction (I&N Chap 13)
Laplace Transform Part 1: Introduction (I&N Chap 13) Definition of the L.T. L.T. of Singularity Functions L.T. Pairs Properties of the L.T. Inverse L.T. Convolution IVT(initial value theorem) & FVT (final
More informationMA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.)
MA 201, Mathematics III, July-November 2018, Laplace Transform (Contd.) Lecture 19 Lecture 19 MA 201, PDE (2018) 1 / 24 Application of Laplace transform in solving ODEs ODEs with constant coefficients
More informationStability. X(s) Y(s) = (s + 2) 2 (s 2) System has 2 poles: points where Y(s) -> at s = +2 and s = -2. Y(s) 8X(s) G 1 G 2
Stability 8X(s) X(s) Y(s) = (s 2) 2 (s 2) System has 2 poles: points where Y(s) -> at s = 2 and s = -2 If all poles are in region where s < 0, system is stable in Fourier language s = jω G 0 - x3 x7 Y(s)
More informationMathQuest: Differential Equations
MathQuest: Differential Equations Laplace Tranforms 1. True or False The Laplace transform method is the only way to solve some types of differential equations. (a) True, and I am very confident (b) True,
More informationName: Solutions Exam 3
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
More informationECE-202 FINAL April 30, 2018 CIRCLE YOUR DIVISION
ECE 202 Final, Spring 8 ECE-202 FINAL April 30, 208 Name: (Please print clearly.) Student Email: CIRCLE YOUR DIVISION DeCarlo- 7:30-8:30 DeCarlo-:30-2:45 2025 202 INSTRUCTIONS There are 34 multiple choice
More information06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1
IV. Continuous-Time Signals & LTI Systems [p. 3] Analog signal definition [p. 4] Periodic signal [p. 5] One-sided signal [p. 6] Finite length signal [p. 7] Impulse function [p. 9] Sampling property [p.11]
More informationLecture 29. Convolution Integrals and Their Applications
Math 245 - Mathematics of Physics and Engineering I Lecture 29. Convolution Integrals and Their Applications March 3, 212 Konstantin Zuev (USC) Math 245, Lecture 29 March 3, 212 1 / 13 Agenda Convolution
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationLaplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France
Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France Laplace Transforms Dr. M. A. A. Shoukat Choudhury 1 Laplace Transforms Important analytical
More informationMAT292 - Calculus III - Fall Solution for Term Test 2 - November 6, 2014 DO NOT WRITE ON THE QR CODE AT THE TOP OF THE PAGES.
MAT9 - Calculus III - Fall 4 Solution for Term Test - November 6, 4 Time allotted: 9 minutes. Aids permitted: None. Full Name: Last First Student ID: Email: @mail.utoronto.ca Instructions DO NOT WRITE
More informationANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)
3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic
More informationINC 341 Feedback Control Systems: Lecture 2 Transfer Function of Dynamic Systems I Asst. Prof. Dr.-Ing. Sudchai Boonto
INC 341 Feedback Control Systems: Lecture 2 Transfer Function of Dynamic Systems I Asst. Prof. Dr.-Ing. Sudchai Boonto Department of Control Systems and Instrumentation Engineering King Mongkut s University
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 5: Calculating the Laplace Transform of a Signal Introduction In this Lecture, you will learn: Laplace Transform of Simple
More informationOrdinary differential equations
Class 11 We will address the following topics Convolution of functions Consider the following question: Suppose that u(t) has Laplace transform U(s), v(t) has Laplace transform V(s), what is the inverse
More informationEE/ME/AE324: Dynamical Systems. Chapter 7: Transform Solutions of Linear Models
EE/ME/AE324: Dynamical Systems Chapter 7: Transform Solutions of Linear Models The Laplace Transform Converts systems or signals from the real time domain, e.g., functions of the real variable t, to the
More information20.6. Transfer Functions. Introduction. Prerequisites. Learning Outcomes
Transfer Functions 2.6 Introduction In this Section we introduce the concept of a transfer function and then use this to obtain a Laplace transform model of a linear engineering system. (A linear engineering
More informationChapter 3 Convolution Representation
Chapter 3 Convolution Representation DT Unit-Impulse Response Consider the DT SISO system: xn [ ] System yn [ ] xn [ ] = δ[ n] If the input signal is and the system has no energy at n = 0, the output yn
More informationHIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method David Levermore Department of Mathematics University of Maryland
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method David Levermore Department of Mathematics University of Maryland 9 December Because the presentation of this material in
More informationEE102 Homework 2, 3, and 4 Solutions
EE12 Prof. S. Boyd EE12 Homework 2, 3, and 4 Solutions 7. Some convolution systems. Consider a convolution system, y(t) = + u(t τ)h(τ) dτ, where h is a function called the kernel or impulse response of
More informationMath 2C03 - Differential Equations. Slides shown in class - Winter Laplace Transforms. March 4, 5, 9, 11, 12, 16,
Math 2C03 - Differential Equations Slides shown in class - Winter 2015 Laplace Transforms March 4, 5, 9, 11, 12, 16, 18... 2015 Laplace Transform used to solve linear ODEs and systems of linear ODEs with
More informationHIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method. David Levermore Department of Mathematics University of Maryland
HIGHER-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS IV: Laplace Transform Method David Levermore Department of Mathematics University of Maryland 6 April Because the presentation of this material in lecture
More informationModule 4. Related web links and videos. 1. FT and ZT
Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link
More information9.2 The Input-Output Description of a System
Lecture Notes on Control Systems/D. Ghose/212 22 9.2 The Input-Output Description of a System The input-output description of a system can be obtained by first defining a delta function, the representation
More informationSolving a RLC Circuit using Convolution with DERIVE for Windows
Solving a RLC Circuit using Convolution with DERIVE for Windows Michel Beaudin École de technologie supérieure, rue Notre-Dame Ouest Montréal (Québec) Canada, H3C K3 mbeaudin@seg.etsmtl.ca - Introduction
More information1 otherwise. Note that the area of the pulse is one. The Dirac delta function (a.k.a. the impulse) can be defined using the pulse as follows:
The Dirac delta function There is a function called the pulse: { if t > Π(t) = 2 otherwise. Note that the area of the pulse is one. The Dirac delta function (a.k.a. the impulse) can be defined using the
More informationIntroduction & Laplace Transforms Lectures 1 & 2
Introduction & Lectures 1 & 2, Professor Department of Electrical and Computer Engineering Colorado State University Fall 2016 Control System Definition of a Control System Group of components that collectively
More informationThe formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d
Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) R(x)
More informationControl Systems. Laplace domain analysis
Control Systems Laplace domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic equations define an Input/Output
More informationOrdinary Differential Equations. Session 7
Ordinary Differential Equations. Session 7 Dr. Marco A Roque Sol 11/16/2018 Laplace Transform Among the tools that are very useful for solving linear differential equations are integral transforms. An
More informationChapter 1 Fundamental Concepts
Chapter 1 Fundamental Concepts 1 Signals A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). These quantities are usually the
More informationSection 6.5 Impulse Functions
Section 6.5 Impulse Functions Key terms/ideas: Unit impulse function (technically a generalized function or distribution ) Dirac delta function Laplace transform of the Dirac delta function IVPs with forcing
More informationControl Systems Engineering (Chapter 2. Modeling in the Frequency Domain) Prof. Kwang-Chun Ho Tel: Fax:
Control Systems Engineering (Chapter 2. Modeling in the Frequency Domain) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 02-760-4253 Fax:02-760-4435 Overview Review on Laplace transform Learn about transfer
More information( ) f (k) = FT (R(x)) = R(k)
Solving ODEs using Fourier Transforms The formulas for derivatives are particularly useful because they reduce ODEs to algebraic expressions. Consider the following ODE d 2 dx + p d 2 dx + q f (x) = R(x)
More informationIndia
italian journal of pure and applied mathematics n. 36 216 (819 826) 819 ANALYTIC SOLUTION FOR RLC CIRCUIT OF NON-INTGR ORDR Jignesh P. Chauhan Department of Applied Mathematics & Humanities S.V. National
More information9.5 The Transfer Function
Lecture Notes on Control Systems/D. Ghose/2012 0 9.5 The Transfer Function Consider the n-th order linear, time-invariant dynamical system. dy a 0 y + a 1 dt + a d 2 y 2 dt + + a d n y 2 n dt b du 0u +
More informationSTABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse
SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 4. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.sigmedia.tv STABILITY Have looked at modeling dynamic systems using differential
More informationOnce again a practical exposition, not fully mathematically rigorous Definition F(s) =
Laplace transforms Once again a practical exposition, not fully mathematically rigorous Definition F(s) = 0 f(t).e -st.dt NB lower limit of integral = 0 unilateral LT more rigorously F(s) = 0 f(t).e -st.dt
More informationThe Laplace Transform and the IVP (Sect. 6.2).
The Laplace Transform and the IVP (Sect..2). Solving differential equations using L ]. Homogeneous IVP. First, second, higher order equations. Non-homogeneous IVP. Recall: Partial fraction decompositions.
More informationTo find the step response of an RC circuit
To find the step response of an RC circuit v( t) v( ) [ v( t) v( )] e tt The time constant = RC The final capacitor voltage v() The initial capacitor voltage v(t ) To find the step response of an RL circuit
More information8 sin 3 V. For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0.
For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0. Spring 2015, Exam #5, Problem #1 4t Answer: e tut 8 sin 3 V 1 For the circuit
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More information20.3. Further Laplace Transforms. Introduction. Prerequisites. Learning Outcomes
Further Laplace Transforms 2.3 Introduction In this Section we introduce the second shift theorem which simplifies the determination of Laplace and inverse Laplace transforms in some complicated cases.
More informationTheory of Linear Systems Exercises. Luigi Palopoli and Daniele Fontanelli
Theory of Linear Systems Exercises Luigi Palopoli and Daniele Fontanelli Dipartimento di Ingegneria e Scienza dell Informazione Università di Trento Contents Chapter. Exercises on the Laplace Transform
More information7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= δ (ω nω 0
Fourier Transform Problems 1. Find the Fourier transform of the following signals: a) f 1 (t )=e 3 t sin(10 t)u (t) b) f 1 (t )=e 4 t cos(10 t)u (t) 2. Find the Fourier transform of the following signals:
More informationDefinition of the Laplace transform. 0 x(t)e st dt
Definition of the Laplace transform Bilateral Laplace Transform: X(s) = x(t)e st dt Unilateral (or one-sided) Laplace Transform: X(s) = 0 x(t)e st dt ECE352 1 Definition of the Laplace transform (cont.)
More informationECE 3620: Laplace Transforms: Chapter 3:
ECE 3620: Laplace Transforms: Chapter 3: 3.1-3.4 Prof. K. Chandra ECE, UMASS Lowell September 21, 2016 1 Analysis of LTI Systems in the Frequency Domain Thus far we have understood the relationship between
More informationAdvanced Engineering Mathematics
Advanced Engineering Mathematics Note 6 Laplace Transforms CHUNG, CHIH-CHUNG Outline Introduction & Partial Fractions Laplace Transform. Linearity. First Shifting Theorem (s-shifting) Transforms of Derivatives
More informationLinear Systems Theory
ME 3253 Linear Systems Theory Review Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system? -- Time domain -- Frequency domain (Laplace
More informationMath Shifting theorems
Math 37 - Shifting theorems Erik Kjær Pedersen November 29, 2005 Let us recall the Dirac delta function. It is a function δ(t) which is 0 everywhere but at t = 0 it is so large that b a (δ(t)dt = when
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces
More informationCHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 5-1 Road Map of the Lecture V Laplace Transform and Transfer
More informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
More informationChapter 6: The Laplace Transform 6.3 Step Functions and
Chapter 6: The Laplace Transform 6.3 Step Functions and Dirac δ 2 April 2018 Step Function Definition: Suppose c is a fixed real number. The unit step function u c is defined as follows: u c (t) = { 0
More informationLaplace Transform Theory - 1
Laplace Transform Theory - 1 Existence of Laplace Transforms Before continuing our use of Laplace transforms for solving DEs, it is worth digressing through a quick investigation of which functions actually
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationFourier series for continuous and discrete time signals
8-9 Signals and Systems Fall 5 Fourier series for continuous and discrete time signals The road to Fourier : Two weeks ago you saw that if we give a complex exponential as an input to a system, the output
More informationLAPLACE TRANSFORMATION AND APPLICATIONS. Laplace transformation It s a transformation method used for solving differential equation.
LAPLACE TRANSFORMATION AND APPLICATIONS Laplace transformation It s a transformation method used for solving differential equation. Advantages The solution of differential equation using LT, progresses
More informationMATH 251 Examination II April 3, 2017 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 3, 2017 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationDESIGN OF CMOS ANALOG INTEGRATED CIRCUITS
DESIGN OF CMOS ANALOG INEGRAED CIRCUIS Franco Maloberti Integrated Microsistems Laboratory University of Pavia Discrete ime Signal Processing F. Maloberti: Design of CMOS Analog Integrated Circuits Discrete
More informationMATH 251 Examination II April 4, 2016 FORM A. Name: Student Number: Section:
MATH 251 Examination II April 4, 2016 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must
More informationHomework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
More information1 Notes on Laplace circuit analysis
Physics - David Kleinfeld - Spring 7 Notes on aplace circuit analysis Background We previously learned that we can transform from the time domain to the frequency domain under steady-state conditions thus
More informationOrdinary Differential Equations
Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International
More informationChapter DEs with Discontinuous Force Functions
Chapter 6 6.4 DEs with Discontinuous Force Functions Discontinuous Force Functions Using Laplace Transform, as in 6.2, we solve nonhomogeneous linear second order DEs with constant coefficients. The only
More informationProblem Set 3: Solution Due on Mon. 7 th Oct. in class. Fall 2013
EE 56: Digital Control Systems Problem Set 3: Solution Due on Mon 7 th Oct in class Fall 23 Problem For the causal LTI system described by the difference equation y k + 2 y k = x k, () (a) By first finding
More informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More informationReview of Fourier Transform
Review of Fourier Transform Fourier series works for periodic signals only. What s about aperiodic signals? This is very large & important class of signals Aperiodic signal can be considered as periodic
More informationSignal and systems. Linear Systems. Luigi Palopoli. Signal and systems p. 1/5
Signal and systems p. 1/5 Signal and systems Linear Systems Luigi Palopoli palopoli@dit.unitn.it Wrap-Up Signal and systems p. 2/5 Signal and systems p. 3/5 Fourier Series We have see that is a signal
More informationCLTI System Response (4A) Young Won Lim 4/11/15
CLTI System Response (4A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationChapter 6: The Laplace Transform. Chih-Wei Liu
Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace
More information