CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang

Size: px
Start display at page:

Download "CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang"

Transcription

1 CHBE3 ECTURE V APACE TRANSFORM AND TRANSFER FUNCTION Profeor Dae Ryook Yang Spring 8 Dept. of Chemical and Biological Engineering 5-

2 Road Map of the ecture V aplace Tranform and Tranfer function Definition of aplace tranform Propertie of aplace tranform Invere aplace tranform Definition of tranfer function How to get the tranfer function Propertie of tranfer function + - Controller Actuator PROCESS Senor ecture IV to VII 5-

3 SOUTION OF INEAR ODE t -order linear ODE Integrating factor: [ xe ] f ( t) e atdt () atdt () High-order linear ODE with contant coeff. Mode: root of characteritic equation For ax ax ax f( t), Depending on the root, mode are pt pt Ditinct root: ( e, e ) pt pt Double root: ( e, te ) Imaginary root: ( t t e co t, e in t) dx For atx ( ) f ( x), I.F. exp( a( t) dt) dt () () xt () [ f() te atdt dtce ] ap apa a( p p)( p p) Many other technique for different cae atdt Solution i a linear combination of mode and the coefficient are decided by the initial condition. 5-3

4 APACE TRANSFORM FOR INEAR ODE AND PDE aplace Tranform Not in time domain, rather in frequency domain Derivative and integral become ome operator. ODE i converted into algebraic equation PDE i converted into ODE in patial coordinate Need invere tranform to recover time-domain olution (D.E. calculation) u(t) ODE or PDE y(t) U() Tranfer Function Y() (Algebraic calculation) 5-4

5 DEFINITION OF APACE TRANSFORM Definition F() i called aplace tranform of f(t). f(t) mut be piecewie continuou. F() contain no information on f(t) for t <. The pat information on f(t) (for t < ) i irrelevant. The i a complex variable called aplace tranform variable Invere aplace tranform and are linear. t F() f () t f () te dt - F f() t ( ) - af () t bf () t af () bf () 5-5

6 APACE TRANSFORM OF FUNCTIONS Contant function, a t a t a a a ae dt e f(t) a t Step function, S(t) for t f() t S() t for t t t S( t) e dt e f(t) t Exponential function, e -bt b b ( ) e e e dt e bt bt t b t f(t) b< b> t 5-6

7 Trigonometric function jt co Euler Identity: e t jint jt jt jt jt int e e cot e e j j j j j jt jt cot e + e j j j t j t int e e j in(t) t Rectangular pule, P(t) for t tw f( t) P( t) h for tw t for t tw t t P( ) t w t he dt e e h t w h f(t) h t w t 5-7

8 Impule function, for t tw f() t () t lim/ tw for tw t tw for t tw t () lim lim t w t e dt e tw t tw w tw f () t f() t 'Hopital' rule: lim lim t gt () t g() t Ramp function, t () t f(t) /t w t w t t t te dt f(t) t t e t e dt e dt t t Integration by part: f ' gdt f g f gdt Refer the Table 3. (Seborg et al.) for other function 5-8

9 5-9

10 5-

11 PROPERTIES OF APACE TRANSFORM Differentiation df t t t f e dt f ( te ) f ( e ) dt (by ibp...) dt t f e dt f() F( ) f() d f t t t t f e dt f () t e f ( ) e dt f e dt f () dt F() f() f() F() f() f() n d f ( n) t ( n) t ( n) t f e dt f() t e f ( ) e dt n dt n ( n) t ( n) d f ( n) f e dt f () f () n dt n n ( n) ( n) () () () () F f f f 5-

12 If f () = f () = f () = = f (n-) () =, Initial condition effect are vanihed. It i very convenient to ue deviation variable o that all the effect of initial condition vanih. df dt d f Tranform of linear differential equation. yt () Y (), ut () U() dt n d f dt n F() () F n F () dy() t Y ( ) (if y() ) dt dy() t yt () Kut () ( y() ) ( ) Y( ) KU( ) dt T T T( ) v ( Tw T) Hv ( H) T ( ) T w( ) t z z H 5-

13 Integration t t ( ) ( ) t f d f d e dt t e t t F( ) f ( ) d f e dt (by ib.. p.) Time delay (Tranlation in time) Derivative of aplace tranform d bt () db t da t f d f b t f a t dt at () dt dt () () eibniz rule: ( ) ( ( )) ( ( )) f t f t t in t () ( )S( ) t ( ) f( t )S( t) f( t) e dt f( ) e d (let t) e f( ) e d e F( ) f(t) t df() d t d t t f e dt f e dt ( t f ) e dt t f ( t) d d d 5-3

14 Final value theorem From the T of differentiation, a approache to zero lim df t e dt lim F ( ) f () dt df dt f ( ) f () lim F ( ) f () f ( ) lim F ( ) dt imitation: f ( ) ha to exit. If it diverge or ocillate, thi theorem i not valid. Initial value theorem From the T of differentiation, a approache to infinity df lim t e dt lim F( ) f() dt df lim e t dt lim F ( ) f () f () lim F ( ) dt 5-4

15 EXAMPE ON APACE TRANSFORM () f(t) 3.5 t for t 3 for t 6 f() t for 6 t 6 t for t f( t).5ts( t).5( t)s( t) 3S( t6) F() f() t ( e ) e 5 For F ( ), find f() and f( ). Uing the initial and final value theorem f() lim F( ) lim f( ) lim F( ) lim 5 5 But the final value theorem i not valid becaue 5 lim f( t) lim e t t t 5-5

16 EXAMPE ON APACE TRANSFORM () What i the final value of the following ytem? xxxin t; x() x() ( )( ) ( )=lim x ( )( ) X () X () X x()= Actually, x( ) cannot be defined due to in t term. Find the aplace tranform for? df() From t f( t) d d tint d ( ) (in t t) 5-6

17 INVERSE APACE TRANSFORM Ued to recover the olution in time domain F() f() t - From the table By partial fraction expanion By inverion uing contour integral Partial fraction expanion t f() t F() e F() d j C - After the partial fraction expanion, it require to know ome imple formula of invere aplace tranform uch a ( n)! e,,,, etc. b n ( ) ( ) 5-7

18 PARTIA FRACTION EXPANSION F () N () N () n D( ) ( p ) ( p ) ( p ) ( p ) n n Cae I: All p i are ditinct and real By a root-finding technique, find all root (time-conuming) Find the coefficient for each fraction Comparion of the coefficient after multiplying the denominator Replace ome value for and olve linear algebraic equation Ue of Heaviide expanion Multiply both ide by a factor, (+p i ), and replace with p i. N () i ( pi) D () Invere T: p f t e e e pt pt n () i n p t 5-8

19 Cae II: Some root are repeated F () N () N () b b D( ) ( p) ( p) ( p) ( p) r r r r r r Each repeated factor have to be eparated firt. Same method a Cae I can be applied. Heaviide expanion for repeated factor r i i! d D( ) () i d N( ) () ( ) r p ( i,, r ) i p Invere T f () t e te t e ( r )! pt pt r r pt 5-9

20 Cae III: Some root are complex F () N () cc ( b) D () d d ( b) Each repeated factor have to be eparated firt. Then, ( b) ( b) ( b) ( b) ( b) Invere T where bd /, d d / 4 c, c b / bt f () t e cot e int bt 5-

21 EXAMPES ON INVERSE APACE TRANSFORM ( 5) A B C D F ( ) (ditinct) ( )( )( 3) 3 Multiply each factor and inert the zero value ( 5) B C D A A5/6 ( )( )( 3) 3 ( 5) A( ) C( ) D( ) B B ( )( 3) 3 ( 5) A( ) B( ) D( ) C C 3/ ( )( 3) 3 ( 5) A( 3) B( 3) C( 3) D D /3 ( )( ) f() t F( ) e e e t t 3 t 5-

22 A B C D F ( ) (repeated) 3 3 ( ) ( ) ( ) ( ) ( A B C)( ) D( ) 3 3 ( ) ( 3 ) ( 3 ) ( ) A D AB D BC D CD AD, AB3D, BC3D, CD A, B, C, D Ue of Heaviide expanion r i ( ) ( ) ( ) ( ) ( i ): 3 d ( i ):! d d ( i ):! d f() t F( ) e te t e e - t t t t () i d N( ) () ( ) r (,, ) i p i r i! d D( ) p 5-

23 ( ) A( ) B CD ( ) (complex) ( 45) ( ) F A B CD ( ) ( )( 4 5) 3 ( A C ) (AB4 CD ) (5C4 D ) 5D AC, AB4CD, 5C4D, 5D A/5, B7/5, C /5, D/5 A ( ) B ( ) 7 B ( ) 5 ( ) 5 ( ) C D () () t co t f t F e t e int t

24 e A B F () ( e ) (Time delay) (4)(3) 4 3 A /(3 ) 4, B /(4 ) 3 /4 / e 3e f() t F() t/4 t/3 ( t)/4 ( t)/3 e e e e S( t) 5-4

25 SOVING ODE BY APACE TRANSFORM Procedure. Given linear ODE with initial condition,. Take aplace tranform and olve for output 3. Invere aplace tranform dy dt Example: Solve for 5 4y ; y() dy 5 +4y 5( Y( ) y()) 4 Y( ) dt 5 (54) Y() 5 Y() (5 4).5.5 yt () Y().5.5e t 5-5

26 TRANSFER FUNCTION () Definition An algebraic expreion for the dynamic relation between the input and output of the proce model dy 5 4 y u; y() U() dt et y y and u u4 Y ().5 (5 4) Y() U() G() U ( ) 54.5 Y () Tranfer Function, G() How to find tranfer function. Find the equilibrium point. If the ytem i nonlinear, then linearize around equil. point 3. Introduce deviation variable 4. Take aplace tranform and olve for output 5. Do the Invere aplace tranform and recover the original variable from deviation variable 5-6

27 TRANSFER FUNCTION () Benefit Once TF i known, the output repone to variou given input can be obtained eaily. yt Y GU G U () () () () () () Interconnected ytem can be analyzed eaily. By block diagram algebra X + - G G3 G Y Y () () () G G X () G() G() G3() Eay to analyze the qualitative behavior of a proce, uch a tability, peed of repone, ocillation, etc. By inpecting Pole and Zero Pole: all atifying D()= Zero: all atifying N()= 5-7

28 TRANSFER FUNCTION (3) Steady-tate Gain: The ratio between ultimate change in input and output ouput ( y( ) y()) Gain= K = input ( u ( ) u ()) For a unit tep change in input, the gain i the change in output Gain may not be definable: for example, integrating procee and procee with utaining ocillation in output From the final value theorem, unit tep change in input with zero initial condition give y( ) K lim Y( ) lim G( ) lim G( ) The tranfer function itelf i an impule repone of the ytem Y () GU () () G () (t) G () 5-8

29 EXAMPE Horizontal cylindrical torage tank (Ex4.7) dm dv qi q dt dt h dv dh V( h) wi( h) dh wi ( h) dt dt wi ( h)/ R ( Rh) ( Rh) h dh dh w i qi q ( qi q) (Nonlinear ODE) dt dt ( D h) h Equilibrium point: ( qi, q, h) ( qi q) /( ( Dh) h) (if qi q, h can be any value in h D.) inearization: dh f f f f ( hq,, q) ( hh) ( qq) ( qq) dt h q q i i i ( hq, i, q) i ( hq, i, q) ( hq, i, q) q i q R w i h 5-9

30 f ( qi q) ( qi q) h h ( D h) h ( hq, i, q) f f, q ( Dh) h q ( Dh) h i ( hq, i, q) ( hq, i, q) H () kqi () kq() H () and Q (): - k k H() and Q i (): et thi term be k Tranfer function between (integrating) Tranfer function between (integrating) h If i near or D, k become very large and i around D/, k become minimum. The model could be quite different depending on the operating condition ued for the linearization. The bet uitable range for the linearization in thi cae i around D/. (le change in gain) inearized model would be valid in very narrow range near. h 5-3

31 PROPERTIES OF TRANSFER FUNCTION Additive property Y Y Y () () () G X G X () () () () X () X () G () G () Y () Y () + + Y() Multiplicative property X () X () X 3 () G () G () X3() G() X() G() G() X() G() G() X() Phyical realizability In a tranfer function, the order of numerator(m) i greater than that of denominator(n): called phyically unrealizable The order of derivative for the input i higher than that of output. (require future input value for current output) 5-3

32 EXAMPES ON TWO TANK SYSTEM Two tank in erie (Ex3.7) No reaction dc V qc qci dt dc V qc qc dt Initial condition: c ()= c ()= kg mol/m 3 (Ue deviation var.) Parameter: V /q= min., V /q=.5 min. Tranfer function () Ci () C () ( V / q) C () ( V / q) C() C() C() Ci() C () Ci() ( V / q) ( V/ q) C C i C C V V q 5-3

33 Pule input P 5.5 C i () ( e ) c i P 5.5 t Equivalent impule input C i ( ) (5.5) ( t).5 Pule repone v. Impule repone 5 C C e () P P.5 () i () ( ) 5 e c t e.5 ( ) P t/ ( ) 5( ) e t.5 C () Ci () () c ( t.5)/ 5( )S(.5).65e t / 5-33

34 5 C C e ( )(.5 ) ( )(.5 ) P P.5 ( ) i ( ) ( ) e.5 c t e e P t/ t/.5 () (5 5 ) C ( ) Ci ( ) ()(.5).5 ()(.5) c.5e.5e t/ t/.5.5 ( ) ( t.5)/ ( t.5)/.5 (5 e 5 e )S( t.5) 5-34

CHE302 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang

CHE302 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang CHE3 ECTURE V APACE TRANSFORM AND TRANSFER FUNCTION Profeor Dae Ryook Yang Fall Dept. of Chemical and Biological Engineering Korea Univerity CHE3 Proce Dynamic and Control Korea Univerity 5- SOUTION OF

More information

CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang

CHBE320 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 information

Practice Problems - Week #7 Laplace - Step Functions, DE Solutions Solutions

Practice Problems - Week #7 Laplace - Step Functions, DE Solutions Solutions For Quetion -6, rewrite the piecewie function uing tep function, ketch their graph, and find F () = Lf(t). 0 0 < t < 2. f(t) = (t 2 4) 2 < t In tep-function form, f(t) = u 2 (t 2 4) The graph i the olid

More information

Correction for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002

Correction for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002 Correction for Simple Sytem Example and Note on Laplace Tranform / Deviation Variable ECHE 55 Fall 22 Conider a tank draining from an initial height of h o at time t =. With no flow into the tank (F in

More information

LAPLACE TRANSFORM REVIEW SOLUTIONS

LAPLACE TRANSFORM REVIEW SOLUTIONS LAPLACE TRANSFORM REVIEW SOLUTIONS. Find the Laplace tranform for the following function. If an image i given, firt write out the function and then take the tranform. a e t inh4t From #8 on the table:

More information

Laplace Transformation

Laplace Transformation Univerity of Technology Electromechanical Department Energy Branch Advance Mathematic Laplace Tranformation nd Cla Lecture 6 Page of 7 Laplace Tranformation Definition Suppoe that f(t) i a piecewie continuou

More information

7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281

7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281 72 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 28 and i 2 Show how Euler formula (page 33) can then be ued to deduce the reult a ( a) 2 b 2 {e at co bt} {e at in bt} b ( a) 2 b 2 5 Under what condition

More information

TMA4125 Matematikk 4N Spring 2016

TMA4125 Matematikk 4N Spring 2016 Norwegian Univerity of Science and Technology Department of Mathematical Science TMA45 Matematikk 4N Spring 6 Solution to problem et 6 In general, unle ele i noted, if f i a function, then F = L(f denote

More information

Chapter 4. The Laplace Transform Method

Chapter 4. The Laplace Transform Method Chapter 4. The Laplace Tranform Method The Laplace Tranform i a tranformation, meaning that it change a function into a new function. Actually, it i a linear tranformation, becaue it convert a linear combination

More information

Digital Control System

Digital Control System Digital Control Sytem Summary # he -tranform play an important role in digital control and dicrete ignal proceing. he -tranform i defined a F () f(k) k () A. Example Conider the following equence: f(k)

More information

Name: Solutions Exam 3

Name: Solutions Exam 3 Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer

More information

ME 375 EXAM #1 Tuesday February 21, 2006

ME 375 EXAM #1 Tuesday February 21, 2006 ME 375 EXAM #1 Tueday February 1, 006 Diviion Adam 11:30 / Savran :30 (circle one) Name Intruction (1) Thi i a cloed book examination, but you are allowed one 8.5x11 crib heet. () You have one hour to

More information

18.03SC Final Exam = x 2 y ( ) + x This problem concerns the differential equation. dy 2

18.03SC Final Exam = x 2 y ( ) + x This problem concerns the differential equation. dy 2 803SC Final Exam Thi problem concern the differential equation dy = x y ( ) dx Let y = f (x) be the olution with f ( ) = 0 (a) Sketch the iocline for lope, 0, and, and ketch the direction field along them

More information

MA 266 FINAL EXAM INSTRUCTIONS May 2, 2005

MA 266 FINAL EXAM INSTRUCTIONS May 2, 2005 MA 66 FINAL EXAM INSTRUCTIONS May, 5 NAME INSTRUCTOR. You mut ue a # pencil on the mark ene heet anwer heet.. If the cover of your quetion booklet i GREEN, write in the TEST/QUIZ NUMBER boxe and blacken

More information

SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5

SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5 SOLUTIONS FOR HOMEWORK SECTION 6.4 AND 6.5 Problem : For each of the following function do the following: (i) Write the function a a piecewie function and ketch it graph, (ii) Write the function a a combination

More information

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get

into a discrete time function. Recall that the table of Laplace/z-transforms is constructed by (i) selecting to get Lecture 25 Introduction to Some Matlab c2d Code in Relation to Sampled Sytem here are many way to convert a continuou time function, { h( t) ; t [0, )} into a dicrete time function { h ( k) ; k {0,,, }}

More information

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions Chapter 4 Laplace Tranform 4 Introduction Reading aignment: In thi chapter we will cover Section 4 45 4 Definition and the Laplace tranform of imple function Given f, a function of time, with value f(t

More information

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions

Reading assignment: In this chapter we will cover Sections Definition and the Laplace transform of simple functions Chapter 4 Laplace Tranform 4 Introduction Reading aignment: In thi chapter we will cover Section 4 45 4 Definition and the Laplace tranform of imple function Given f, a function of time, with value f(t

More information

Laplace Transforms Chapter 3

Laplace 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 information

Name: Solutions Exam 2

Name: Solutions Exam 2 Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer

More information

Design of Digital Filters

Design of Digital Filters Deign of Digital Filter Paley-Wiener Theorem [ ] ( ) If h n i a caual energy ignal, then ln H e dω< B where B i a finite upper bound. One implication of the Paley-Wiener theorem i that a tranfer function

More information

Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is

Given the following circuit with unknown initial capacitor voltage v(0): X(s) Immediately, we know that the transfer function H(s) is EE 4G Note: Chapter 6 Intructor: Cheung More about ZSR and ZIR. Finding unknown initial condition: Given the following circuit with unknown initial capacitor voltage v0: F v0/ / Input xt 0Ω Output yt -

More information

ME 375 FINAL EXAM Wednesday, May 6, 2009

ME 375 FINAL EXAM Wednesday, May 6, 2009 ME 375 FINAL EXAM Wedneday, May 6, 9 Diviion Meckl :3 / Adam :3 (circle one) Name_ Intruction () Thi i a cloed book examination, but you are allowed three ingle-ided 8.5 crib heet. A calculator i NOT allowed.

More information

SECTION x2 x > 0, t > 0, (8.19a)

SECTION x2 x > 0, t > 0, (8.19a) SECTION 8.5 433 8.5 Application of aplace Tranform to Partial Differential Equation In Section 8.2 and 8.3 we illutrated the effective ue of aplace tranform in olving ordinary differential equation. The

More information

Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas)

Lecture 17: Analytic Functions and Integrals (See Chapter 14 in Boas) Lecture 7: Analytic Function and Integral (See Chapter 4 in Boa) Thi i a good point to take a brief detour and expand on our previou dicuion of complex variable and complex function of complex variable.

More information

1 Routh Array: 15 points

1 Routh Array: 15 points EE C28 / ME34 Problem Set 3 Solution Fall 2 Routh Array: 5 point Conider the ytem below, with D() k(+), w(t), G() +2, and H y() 2 ++2 2(+). Find the cloed loop tranfer function Y () R(), and range of k

More information

Linear System Fundamentals

Linear System Fundamentals Linear Sytem Fundamental MEM 355 Performance Enhancement of Dynamical Sytem Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Content Sytem Repreentation Stability Concept

More information

Solving Differential Equations by the Laplace Transform and by Numerical Methods

Solving Differential Equations by the Laplace Transform and by Numerical Methods 36CH_PHCalter_TechMath_95099 3//007 :8 PM Page Solving Differential Equation by the Laplace Tranform and by Numerical Method OBJECTIVES When you have completed thi chapter, you hould be able to: Find the

More information

Name: Solutions Exam 2

Name: Solutions Exam 2 Name: Solution Exam Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will

More information

e st t u(t 2) dt = lim t dt = T 2 2 e st = T e st lim + e st

e st t u(t 2) dt = lim t dt = T 2 2 e st = T e st lim + e st Math 46, Profeor David Levermore Anwer to Quetion for Dicuion Friday, 7 October 7 Firt Set of Quetion ( Ue the definition of the Laplace tranform to compute Lf]( for the function f(t = u(t t, where u i

More information

Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:

Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web:     Ph: Serial :. PT_EE_A+C_Control Sytem_798 Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubanewar olkata Patna Web: E-mail: info@madeeay.in Ph: -4546 CLASS TEST 8-9 ELECTRICAL ENGINEERING Subject

More information

Lecture 6: Resonance II. Announcements

Lecture 6: Resonance II. Announcements EES 5 Spring 4, Lecture 6 Lecture 6: Reonance II EES 5 Spring 4, Lecture 6 Announcement The lab tart thi week You mut how up for lab to tay enrolled in the coure. The firt lab i available on the web ite,

More information

Chapter 7: The Laplace Transform Part 1

Chapter 7: The Laplace Transform Part 1 Chapter 7: The Laplace Tranform Part 1 王奕翔 Department of Electrical Engineering National Taiwan Univerity ihwang@ntu.edu.tw November 26, 213 1 / 34 王奕翔 DE Lecture 1 Solving an initial value problem aociated

More information

CONTROL SYSTEMS. Chapter 2 : Block Diagram & Signal Flow Graphs GATE Objective & Numerical Type Questions

CONTROL SYSTEMS. Chapter 2 : Block Diagram & Signal Flow Graphs GATE Objective & Numerical Type Questions ONTOL SYSTEMS hapter : Bloc Diagram & Signal Flow Graph GATE Objective & Numerical Type Quetion Quetion 6 [Practice Boo] [GATE E 994 IIT-Kharagpur : 5 Mar] educe the ignal flow graph hown in figure below,

More information

MODERN CONTROL SYSTEMS

MODERN CONTROL SYSTEMS MODERN CONTROL SYSTEMS Lecture 1 Root Locu Emam Fathy Department of Electrical and Control Engineering email: emfmz@aat.edu http://www.aat.edu/cv.php?dip_unit=346&er=68525 1 Introduction What i root locu?

More information

Control Systems Engineering ( Chapter 7. Steady-State Errors ) Prof. Kwang-Chun Ho Tel: Fax:

Control Systems Engineering ( Chapter 7. Steady-State Errors ) Prof. Kwang-Chun Ho Tel: Fax: Control Sytem Engineering ( Chapter 7. Steady-State Error Prof. Kwang-Chun Ho kwangho@hanung.ac.kr Tel: 0-760-453 Fax:0-760-4435 Introduction In thi leon, you will learn the following : How to find the

More information

Math 201 Lecture 17: Discontinuous and Periodic Functions

Math 201 Lecture 17: Discontinuous and Periodic Functions Math 2 Lecture 7: Dicontinuou and Periodic Function Feb. 5, 22 Many example here are taken from the textbook. he firt number in () refer to the problem number in the UA Cutom edition, the econd number

More information

EE105 - Fall 2005 Microelectronic Devices and Circuits

EE105 - Fall 2005 Microelectronic Devices and Circuits EE5 - Fall 5 Microelectronic Device and ircuit Lecture 9 Second-Order ircuit Amplifier Frequency Repone Announcement Homework 8 due tomorrow noon Lab 7 next week Reading: hapter.,.3. Lecture Material Lat

More information

MAHALAKSHMI ENGINEERING COLLEGE-TRICHY

MAHALAKSHMI ENGINEERING COLLEGE-TRICHY DIGITAL SIGNAL PROCESSING DEPT./SEM.: CSE /VII DIGITAL FILTER DESIGN-IIR & FIR FILTER DESIGN PART-A. Lit the different type of tructure for realiation of IIR ytem? AUC APR 09 The different type of tructure

More information

The Laplace Transform

The Laplace Transform The Laplace Tranform Prof. Siripong Potiuk Pierre Simon De Laplace 749-827 French Atronomer and Mathematician Laplace Tranform An extenion of the CT Fourier tranform to allow analyi of broader cla of CT

More information

These are practice problems for the final exam. You should attempt all of them, but turn in only the even-numbered problems!

These are practice problems for the final exam. You should attempt all of them, but turn in only the even-numbered problems! Math 33 - ODE Due: 7 December 208 Written Problem Set # 4 Thee are practice problem for the final exam. You hould attempt all of them, but turn in only the even-numbered problem! Exercie Solve the initial

More information

Laplace Transforms. Chapter 3. Pierre Simon Laplace Born: 23 March 1749 in Beaumont-en-Auge, Normandy, France Died: 5 March 1827 in Paris, France

Laplace 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 information

The Riemann Transform

The Riemann Transform The Riemann Tranform By Armando M. Evangelita Jr. armando78973@gmail.com Augut 28, 28 ABSTRACT In hi 859 paper, Bernhard Riemann ued the integral equation f (x ) x dx to develop an explicit formula for

More information

Midterm Test Nov 10, 2010 Student Number:

Midterm Test Nov 10, 2010 Student Number: Mathematic 265 Section: 03 Verion A Full Name: Midterm Tet Nov 0, 200 Student Number: Intruction: There are 6 page in thi tet (including thi cover page).. Caution: There may (or may not) be more than one

More information

Definitions of the Laplace Transform (1A) Young Won Lim 2/9/15

Definitions of the Laplace Transform (1A) Young Won Lim 2/9/15 Definition of the aplace Tranform (A) 2/9/5 Copyright (c) 24 Young W. im. Permiion i granted to copy, ditriute and/or modify thi document under the term of the GNU Free Documentation icene, Verion.2 or

More information

ME2142/ME2142E Feedback Control Systems

ME2142/ME2142E Feedback Control Systems Root Locu Analyi Root Locu Analyi Conider the cloed-loop ytem R + E - G C B H The tranient repone, and tability, of the cloed-loop ytem i determined by the value of the root of the characteritic equation

More information

EXAM 4 -B2 MATH 261: Elementary Differential Equations MATH 261 FALL 2012 EXAMINATION COVER PAGE Professor Moseley

EXAM 4 -B2 MATH 261: Elementary Differential Equations MATH 261 FALL 2012 EXAMINATION COVER PAGE Professor Moseley EXAM 4 -B MATH 6: Elementary Differential Equation MATH 6 FALL 0 EXAMINATION COVER PAGE Profeor Moeley PRINT NAME ( ) Lat Name, Firt Name MI (What you wih to be called) ID # EXAM DATE Friday, Nov. 9, 0

More information

MATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.:

MATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.: MATEMATIK Datum: 20-08-25 Tid: eftermiddag GU, Chalmer Hjälpmedel: inga A.Heintz Telefonvakt: Ander Martinon Tel.: 073-07926. Löningar till tenta i ODE och matematik modellering, MMG5, MVE6. Define what

More information

Introduction to Laplace Transform Techniques in Circuit Analysis

Introduction to Laplace Transform Techniques in Circuit Analysis Unit 6 Introduction to Laplace Tranform Technique in Circuit Analyi In thi unit we conider the application of Laplace Tranform to circuit analyi. A relevant dicuion of the one-ided Laplace tranform i found

More information

L 2 -transforms for boundary value problems

L 2 -transforms for boundary value problems Computational Method for Differential Equation http://cmde.tabrizu.ac.ir Vol. 6, No., 8, pp. 76-85 L -tranform for boundary value problem Arman Aghili Department of applied mathematic, faculty of mathematical

More information

Chapter 7: The Laplace Transform

Chapter 7: The Laplace Transform Chapter 7: The Laplace Tranform 王奕翔 Department of Electrical Engineering National Taiwan Univerity ihwang@ntu.edu.tw November 2, 213 1 / 25 王奕翔 DE Lecture 1 Solving an initial value problem aociated with

More information

EXAM 4 -A2 MATH 261: Elementary Differential Equations MATH 261 FALL 2010 EXAMINATION COVER PAGE Professor Moseley

EXAM 4 -A2 MATH 261: Elementary Differential Equations MATH 261 FALL 2010 EXAMINATION COVER PAGE Professor Moseley EXAM 4 -A MATH 6: Elementary Differential Equation MATH 6 FALL 00 EXAMINATION COVER PAGE Profeor Moeley PRINT NAME ( ) Lat Name, Firt Name MI (What you wih to be called) ID # EXAM DATE Friday, Nov. 9,

More information

The Laplace Transform

The Laplace Transform Chapter 7 The Laplace Tranform 85 In thi chapter we will explore a method for olving linear differential equation with contant coefficient that i widely ued in electrical engineering. It involve the tranformation

More information

Question 1 Equivalent Circuits

Question 1 Equivalent Circuits MAE 40 inear ircuit Fall 2007 Final Intruction ) Thi exam i open book You may ue whatever written material you chooe, including your cla note and textbook You may ue a hand calculator with no communication

More information

ECE382/ME482 Spring 2004 Homework 4 Solution November 14,

ECE382/ME482 Spring 2004 Homework 4 Solution November 14, ECE382/ME482 Spring 2004 Homework 4 Solution November 14, 2005 1 Solution to HW4 AP4.3 Intead of a contant or tep reference input, we are given, in thi problem, a more complicated reference path, r(t)

More information

MATH 251 Examination II April 6, 2015 FORM A. Name: Student Number: Section:

MATH 251 Examination II April 6, 2015 FORM A. Name: Student Number: Section: MATH 251 Examination II April 6, 2015 FORM A Name: Student Number: Section: Thi exam ha 12 quetion for a total of 100 point. In order to obtain full credit for partial credit problem, all work mut be hown.

More information

Exercises for lectures 19 Polynomial methods

Exercises for lectures 19 Polynomial methods Exercie for lecture 19 Polynomial method Michael Šebek Automatic control 016 15-4-17 Diviion of polynomial with and without remainder Polynomial form a circle, but not a body. (Circle alo form integer,

More information

ME 375 FINAL EXAM SOLUTIONS Friday December 17, 2004

ME 375 FINAL EXAM SOLUTIONS Friday December 17, 2004 ME 375 FINAL EXAM SOLUTIONS Friday December 7, 004 Diviion Adam 0:30 / Yao :30 (circle one) Name Intruction () Thi i a cloed book eamination, but you are allowed three 8.5 crib heet. () You have two hour

More information

Equivalent POG block schemes

Equivalent POG block schemes apitolo. NTRODUTON 3. Equivalent OG block cheme et u conider the following inductor connected in erie: 2 Three mathematically equivalent OG block cheme can be ued: a) nitial condition φ φ b) nitial condition

More information

Chapter 13. Root Locus Introduction

Chapter 13. Root Locus Introduction Chapter 13 Root Locu 13.1 Introduction In the previou chapter we had a glimpe of controller deign iue through ome imple example. Obviouly when we have higher order ytem, uch imple deign technique will

More information

The Power Series Expansion on a Bulge Heaviside Step Function

The Power Series Expansion on a Bulge Heaviside Step Function Applied Mathematical Science, Vol 9, 05, no 3, 5-9 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/0988/am054009 The Power Serie Expanion on a Bulge Heaviide Step Function P Haara and S Pothat Department of

More information

Chapter 7. Root Locus Analysis

Chapter 7. Root Locus Analysis Chapter 7 Root Locu Analyi jw + KGH ( ) GH ( ) - K 0 z O 4 p 2 p 3 p Root Locu Analyi The root of the cloed-loop characteritic equation define the ytem characteritic repone. Their location in the complex

More information

Basic Procedures for Common Problems

Basic 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 information

LAPLACE TRANSFORM AND TRANSFER FUNCTION

LAPLACE 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 information

Bogoliubov Transformation in Classical Mechanics

Bogoliubov Transformation in Classical Mechanics Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How

More information

( 1) EE 313 Linear Signals & Systems (Fall 2018) Solution Set for Homework #10 on Laplace Transforms

( 1) EE 313 Linear Signals & Systems (Fall 2018) Solution Set for Homework #10 on Laplace Transforms EE 33 Linear Signal & Sytem (Fall 08) Solution Set for Homework #0 on Laplace Tranform By: Mr. Houhang Salimian & Prof. Brian L. Evan Problem. a) xt () = ut () ut ( ) From lecture Lut { ()} = and { } t

More information

Figure 1: Unity Feedback System

Figure 1: Unity Feedback System MEM 355 Sample Midterm Problem Stability 1 a) I the following ytem table? Solution: G() = Pole: -1, -2, -2, -1.5000 + 1.3229i, -1.5000-1.3229i 1 ( + 1)( 2 + 3 + 4)( + 2) 2 A you can ee, all pole are on

More information

EE Control Systems LECTURE 6

EE Control Systems LECTURE 6 Copyright FL Lewi 999 All right reerved EE - Control Sytem LECTURE 6 Updated: Sunday, February, 999 BLOCK DIAGRAM AND MASON'S FORMULA A linear time-invariant (LTI) ytem can be repreented in many way, including:

More information

The Laplace Transform , Haynes Miller and Jeremy Orloff

The Laplace Transform , Haynes Miller and Jeremy Orloff The Laplace Tranform 8.3, Hayne Miller and Jeremy Orloff Laplace tranform baic: introduction An operator take a function a input and output another function. A tranform doe the ame thing with the added

More information

Wolfgang Hofle. CERN CAS Darmstadt, October W. Hofle feedback systems

Wolfgang Hofle. CERN CAS Darmstadt, October W. Hofle feedback systems Wolfgang Hofle Wolfgang.Hofle@cern.ch CERN CAS Darmtadt, October 9 Feedback i a mechanim that influence a ytem by looping back an output to the input a concept which i found in abundance in nature and

More information

The state variable description of an LTI system is given by 3 1O. Statement for Linked Answer Questions 3 and 4 :

The state variable description of an LTI system is given by 3 1O. Statement for Linked Answer Questions 3 and 4 : CHAPTER 6 CONTROL SYSTEMS YEAR TO MARKS MCQ 6. The tate variable decription of an LTI ytem i given by Jxo N J a NJx N JN K O K OK O K O xo a x + u Kxo O K 3 a3 OKx O K 3 O L P L J PL P L P x N K O y _

More information

R L R L L sl C L 1 sc

R L R L L sl C L 1 sc 2260 N. Cotter PRACTICE FINAL EXAM SOLUTION: Prob 3 3. (50 point) u(t) V i(t) L - R v(t) C - The initial energy tored in the circuit i zero. 500 Ω L 200 mh a. Chooe value of R and C to accomplih the following:

More information

EECS2200 Electric Circuits. RLC Circuit Natural and Step Responses

EECS2200 Electric Circuits. RLC Circuit Natural and Step Responses 5--4 EECS Electric Circuit Chapter 6 R Circuit Natural and Step Repone Objective Determine the repone form of the circuit Natural repone parallel R circuit Natural repone erie R circuit Step repone of

More information

Modeling in the Frequency Domain

Modeling in the Frequency Domain T W O Modeling in the Frequency Domain SOLUTIONS TO CASE STUDIES CHALLENGES Antenna Control: Tranfer Function Finding each tranfer function: Pot: V i θ i 0 π ; Pre-Amp: V p V i K; Power Amp: E a V p 50

More information

7-5-S-Laplace Transform Issues and Opportunities in Mathematica

7-5-S-Laplace Transform Issues and Opportunities in Mathematica 7-5-S-Laplace Tranform Iue and Opportunitie in Mathematica The Laplace Tranform i a mathematical contruct that ha proven very ueful in both olving and undertanding differential equation. We introduce it

More information

CHAPTER 9. Inverse Transform and. Solution to the Initial Value Problem

CHAPTER 9. Inverse Transform and. Solution to the Initial Value Problem A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES A COLLECTION OF HANDOUTS ON SCALAR LINEAR ORDINARY DIFFERENTIAL

More information

Root Locus Diagram. Root loci: The portion of root locus when k assume positive values: that is 0

Root Locus Diagram. Root loci: The portion of root locus when k assume positive values: that is 0 Objective Root Locu Diagram Upon completion of thi chapter you will be able to: Plot the Root Locu for a given Tranfer Function by varying gain of the ytem, Analye the tability of the ytem from the root

More information

SIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuits II. R 4 := 100 kohm

SIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuits II. R 4 := 100 kohm SIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuit II Solution to Aignment 3 February 2003. Cacaded Op Amp [DC&L, problem 4.29] An ideal op amp ha an output impedance of zero,

More information

Function and Impulse Response

Function and Impulse Response Tranfer Function and Impule Repone Solution of Selected Unolved Example. Tranfer Function Q.8 Solution : The -domain network i hown in the Fig... Applying VL to the two loop, R R R I () I () L I () L V()

More information

Chapter 10. Closed-Loop Control Systems

Chapter 10. Closed-Loop Control Systems hapter 0 loed-loop ontrol Sytem ontrol Diagram of a Typical ontrol Loop Actuator Sytem F F 2 T T 2 ontroller T Senor Sytem T TT omponent and Signal of a Typical ontrol Loop F F 2 T Air 3-5 pig 4-20 ma

More information

8. [12 Points] Find a particular solution of the differential equation. t 2 y + ty 4y = t 3, y h = c 1 t 2 + c 2 t 2.

8. [12 Points] Find a particular solution of the differential equation. t 2 y + ty 4y = t 3, y h = c 1 t 2 + c 2 t 2. Intruction. Anwer each of the quetion on your own paper. Put your name on each page of your paper. Be ure to how your work o that partial credit can be adequately aeed. Credit will not be given for anwer

More information

R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder

R. W. Erickson. Department of Electrical, Computer, and Energy Engineering University of Colorado, Boulder R. W. Erickon Department of Electrical, Computer, and Energy Engineering Univerity of Colorado, Boulder Cloed-loop buck converter example: Section 9.5.4 In ECEN 5797, we ued the CCM mall ignal model to

More information

4e st dt. 0 e st dt. lim. f (t)e st dt. f (t) e st dt + 0. e (2 s)t dt + 0. e (2 s)4 1 = 1. = t s e st

4e st dt. 0 e st dt. lim. f (t)e st dt. f (t) e st dt + 0. e (2 s)t dt + 0. e (2 s)4 1 = 1. = t s e st Worked Solution 8 Chapter : The Laplace Tranform 6 a F L] e t dt e t dt e t ] lim t e t e if > for > 6 c F L f t] f te t dt f t e t dt + e t dt + e t + f t e t dt e t dt ] e e ] 6 e F L f t] f te t dt

More information

Solutions to homework #10

Solutions to homework #10 Solution to homework #0 Problem 7..3 Compute 6 e 3 t t t 8. The firt tep i to ue the linearity of the Laplace tranform to ditribute the tranform over the um and pull the contant factor outide the tranform.

More information

Math 334 Fall 2011 Homework 10 Solutions

Math 334 Fall 2011 Homework 10 Solutions Nov. 5, Math 334 Fall Homework Solution Baic Problem. Expre the following function uing the unit tep function. And ketch their graph. < t < a g(t = < t < t > t t < b g(t = t Solution. a We

More information

ANSWERS TO MA1506 TUTORIAL 7. Question 1. (a) We shall use the following s-shifting property: L(f(t)) = F (s) L(e ct f(t)) = F (s c)

ANSWERS TO MA1506 TUTORIAL 7. Question 1. (a) We shall use the following s-shifting property: L(f(t)) = F (s) L(e ct f(t)) = F (s c) ANSWERS O MA56 UORIAL 7 Quetion. a) We hall ue the following -Shifting property: Lft)) = F ) Le ct ft)) = F c) Lt 2 ) = 2 3 ue Ltn ) = n! Lt 2 e 3t ) = Le 3t t 2 ) = n+ 2 + 3) 3 b) Here u denote the Unit

More information

Things to Definitely Know. e iθ = cos θ + i sin θ. cos 2 θ + sin 2 θ = 1. cos(u + v) = cos u cos v sin u sin v sin(u + v) = cos u sin v + sin u cos v

Things to Definitely Know. e iθ = cos θ + i sin θ. cos 2 θ + sin 2 θ = 1. cos(u + v) = cos u cos v sin u sin v sin(u + v) = cos u sin v + sin u cos v Thing to Definitely Know Euler Identity Pythagorean Identity Trigonometric Identitie e iθ co θ + i in θ co 2 θ + in 2 θ I Firt Order Differential Equation co(u + v co u co v in u in v in(u + v co u in

More information

Analysis of Step Response, Impulse and Ramp Response in the Continuous Stirred Tank Reactor System

Analysis of Step Response, Impulse and Ramp Response in the Continuous Stirred Tank Reactor System ISSN: 454-50 Volume 0 - Iue 05 May 07 PP. 7-78 Analyi of Step Repone, Impule and Ramp Repone in the ontinuou Stirred Tank Reactor Sytem * Zohreh Khohraftar, Pirouz Derakhhi, (Department of hemitry, Science

More information

Math 273 Solutions to Review Problems for Exam 1

Math 273 Solutions to Review Problems for Exam 1 Math 7 Solution to Review Problem for Exam True or Fale? Circle ONE anwer for each Hint: For effective tudy, explain why if true and give a counterexample if fale (a) T or F : If a b and b c, then a c

More information

Sampling and the Discrete Fourier Transform

Sampling and the Discrete Fourier Transform Sampling and the Dicrete Fourier Tranform Sampling Method Sampling i mot commonly done with two device, the ample-and-hold (S/H) and the analog-to-digital-converter (ADC) The S/H acquire a CT ignal at

More information

Lecture 4. Chapter 11 Nise. Controller Design via Frequency Response. G. Hovland 2004

Lecture 4. Chapter 11 Nise. Controller Design via Frequency Response. G. Hovland 2004 METR4200 Advanced Control Lecture 4 Chapter Nie Controller Deign via Frequency Repone G. Hovland 2004 Deign Goal Tranient repone via imple gain adjutment Cacade compenator to improve teady-tate error Cacade

More information

Module 4: Time Response of discrete time systems Lecture Note 1

Module 4: Time Response of discrete time systems Lecture Note 1 Digital Control Module 4 Lecture Module 4: ime Repone of dicrete time ytem Lecture Note ime Repone of dicrete time ytem Abolute tability i a baic requirement of all control ytem. Apart from that, good

More information

March 18, 2014 Academic Year 2013/14

March 18, 2014 Academic Year 2013/14 POLITONG - SHANGHAI BASIC AUTOMATIC CONTROL Exam grade March 8, 4 Academic Year 3/4 NAME (Pinyin/Italian)... STUDENT ID Ue only thee page (including the back) for anwer. Do not ue additional heet. Ue of

More information

Digital Control System

Digital Control System Digital Control Sytem - A D D A Micro ADC DAC Proceor Correction Element Proce Clock Meaurement A: Analog D: Digital Continuou Controller and Digital Control Rt - c Plant yt Continuou Controller Digital

More information

Problem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that

Problem 1. Construct a filtered probability space on which a Brownian motion W and an adapted process X are defined and such that Stochatic Calculu Example heet 4 - Lent 5 Michael Tehranchi Problem. Contruct a filtered probability pace on which a Brownian motion W and an adapted proce X are defined and uch that dx t = X t t dt +

More information

IEOR 3106: Fall 2013, Professor Whitt Topics for Discussion: Tuesday, November 19 Alternating Renewal Processes and The Renewal Equation

IEOR 3106: Fall 2013, Professor Whitt Topics for Discussion: Tuesday, November 19 Alternating Renewal Processes and The Renewal Equation IEOR 316: Fall 213, Profeor Whitt Topic for Dicuion: Tueday, November 19 Alternating Renewal Procee and The Renewal Equation 1 Alternating Renewal Procee An alternating renewal proce alternate between

More information

Main Topics: The Past, H(s): Poles, zeros, s-plane, and stability; Decomposition of the complete response.

Main Topics: The Past, H(s): Poles, zeros, s-plane, and stability; Decomposition of the complete response. EE202 HOMEWORK PROBLEMS SPRING 18 TO THE STUDENT: ALWAYS CHECK THE ERRATA on the web. Quote for your Parent' Partie: 1. Only with nodal analyi i the ret of the emeter a poibility. Ray DeCarlo 2. (The need

More information

( ) ( ) ω = X x t e dt

( ) ( ) ω = X x t e dt The Laplace Tranform The Laplace Tranform generalize the Fourier Traform for the entire complex plane For an ignal x(t) the pectrum, or it Fourier tranform i (if it exit): t X x t e dt ω = For the ame

More information

Chapter 2: Problem Solutions

Chapter 2: Problem Solutions Chapter 2: Solution Dicrete Time Proceing of Continuou Time Signal Sampling à 2.. : Conider a inuoidal ignal and let u ample it at a frequency F 2kHz. xt 3co000t 0. a) Determine and expreion for the ampled

More information

NAME (pinyin/italian)... MATRICULATION NUMBER... SIGNATURE

NAME (pinyin/italian)... MATRICULATION NUMBER... SIGNATURE POLITONG SHANGHAI BASIC AUTOMATIC CONTROL June Academic Year / Exam grade NAME (pinyin/italian)... MATRICULATION NUMBER... SIGNATURE Ue only thee page (including the bac) for anwer. Do not ue additional

More information