Solutions to homework #10
|
|
- Oscar Horace Sanders
- 5 years ago
- Views:
Transcription
1 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. That give 6 e 3 t t t 8 6 e 3 t t t 8. Uing the book table, we find e 3 t, 3 (entry 4 with a 3) t, (entry 5 with n ) 3 t, (entry 5 with n ) and Therefore 6 e 3 t t 6 t 8 3 (entry 4 with a 0) Calculation with Mathematica We can alo compute the Laplace tranform automatically uing Mathematica, In[]:= LaplaceTranform 6 Exp 3 t t^ t 8, t, Out[]= which i identical to the reult we got uing table lookup.
2 hwk0soln.nb Problem 7..5 Compute t 3 t e t e 4 t co t. 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. That give t 3 t e t e 4 t co t t 3 t e t e 4 t co t. Uing the book table, we find 6 t 3, (entry 5 with n 3) 4 t e t, (entry 5 with n and a ) and e 4 t 4 co t 4 Therefore t 3 t e t e 4 t 6 co t 4. (entry 6 with a 4 and b ). 4 4 Calculation with Mathematica We can alo compute the Laplace tranform automatically uing Mathematica, In[3]:= LaplaceTranform t^3 t Exp t Exp 4 t Co t, t, Out[3]= which i identical to the reult we got uing table lookup. Problem Find the invere Laplace tranform of F 0 0 a
3 hwk0soln.nb 3 b^ 4 a c mut be poitive. Computing the dicriminant with a=, b=, and c=0 give the value 36, o the denominator doen t factor in real number. Therefore we proceed to complete the quare. Recall that to complete the quare we write b c in the form A B. Expanding the quare give A A B. We then equate coefficient of each power of (which we can do becaue the power of are linearly independent function) to find the equation and b A c A B. With b and c 0 we find A and B 3. To check a completion of quare, plug A and B into A B and expand it out. You might want to try it by hand; it alo eay to do in Mathematica: Expand 3 0 which i the original polynomial. Therefore we ve completed the quare correctly, and we can rewrite F a. 3 Thi i entry #6 in the table on the book back cover, o we can recognize it a the Laplace tranform of f t e t co 3 t. Calculation in Mathematica I hould alo point out that you can do the whole work in Mathematica uing the InvereLaplaceTranform function: ILT t InvereLaplaceTranform F,, t 3 t 6 t f t FullSimplify ILT t t co 3 t which i identical to the reult found by algebraic manipulation and table lookup. Finally, we can check by computing the Laplace tranform of the anwer LaplaceTranform Exp t Co 3 t, t, 9 Together ExpandAll 0 which i the F with which we tarted. Therefore our calculation check.
4 4 hwk0soln.nb Problem Find the invere Laplace tranform of F The denominator doen t factor in the real (which you can check) o we complete the quare. Uing the procedure for completing the quare hown in the previou problem, the denominator can be rewritten a 4 (which you hould alo check). Looking up F 4 4 Thi i cloe to entry #5 in the table; it differ by a factor of Check by computing the Laplace tranform: LaplaceTranform Exp t Sin t, t, 4 ExpandAll 4 8 which recover the tarting point. Therefore the calculation check. Thu the invere Laplace tranform of F i f t e t in t. Calculation with Mathematica Alternatively, we can ue the InvereLaplaceTranform function to do the problem in one hot: ILT t InvereLaplaceTranform ^ 4 8,, t 4 t 4 t FullSimplify ILT t in t coh t inh t Thi look different, but uing the definition of the hyperbolic function you can how that it i the ame a the reult we got uing table lookup.
5 hwk0soln.nb 5 Problem Table lookup & hand calculation Solve for Y y given y 3 y y co t with initial condition y 0 0 and y 0. Conulting our friendly table, we find y Y y 0 y 0 (entry 4) y Y y 0 (entry 3) o that the Laplace tranform of the LHS i y 3 y y 3 Y. We can alo look up the Laplace tranform of the RHS,. co t Therefore 3 Y from which we can olve for Y, Y. 3 3 Adding the two term give Y. 3 Calculation with Mathematica Here how to do it emi-automatically. Firt compute the Laplace tranform of the LHS In[5]:= Out[5]= LHS LaplaceTranform y t 3 y t y t, t, t y t y 0 t y t 3 t y t y 0 y 0 Becaue Mathematica doen t yet know the initial condition we have to ue tranformation rule to replace y 0 and y 0 with the initial value. Furthermore, the Laplace tranform of the unpecified function y t can t be computed; calling the LaplaceTranform function on it imply return a "placeholder" expreion: In[9]:= LaplaceTranform y t, t, Out[9]= t y t Therefore I ll alo ue a tranformation rule to replace LaplaceTranform[y[t], t, ] with Y. In Mathematica, thi i done a follow: In[6]:= LHS LHS. y 0, y 0 0, LaplaceTranform y t, t, Y reulting in Out[6]= Y 3 Y Y Now I take the Laplace tranform of the RHS In[7]:= Out[7]= RHS LaplaceTranform Co t, t,
6 6 hwk0soln.nb Having computed the Laplace tranform of both ide, we can et up an equation for Y, In[0]:= eqn LHS RHS Out[0]= Y 3 Y Y which we can olve for Y In[8]:= Out[8]= Solve LHS RHS, Y Y 3 Thi i identical to the reult computed by hand.
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 informationLaplace 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 informationPractice 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 informationSolving 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 informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Laplace Tranform Paul Dawkin Table of Content Preface... Laplace Tranform... Introduction... The Definition... 5 Laplace Tranform... 9 Invere Laplace Tranform... Step Function...4
More informationMoment of Inertia of an Equilateral Triangle with Pivot at one Vertex
oment of nertia of an Equilateral Triangle with Pivot at one Vertex There are two wa (at leat) to derive the expreion f an equilateral triangle that i rotated about one vertex, and ll how ou both here.
More informationDIFFERENTIAL EQUATIONS Laplace Transforms. Paul Dawkins
DIFFERENTIAL EQUATIONS Laplace Tranform Paul Dawkin Table of Content Preface... Laplace Tranform... Introduction... The Definition... 5 Laplace Tranform... 9 Invere Laplace Tranform... Step Function...
More informationME 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 information7.2 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 informationR 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 informationSolutions for homework 8
Solution for homework 8 Section. Baic propertie of the Laplace Tranform. Ue the linearity of the Laplace tranform (Propoition.7) and Table of Laplace tranform on page 04 to find the Laplace tranform of
More informatione 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 informationLAPLACE 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 informationHomework #7 Solution. Solutions: ΔP L Δω. Fig. 1
Homework #7 Solution Aignment:. through.6 Bergen & Vittal. M Solution: Modified Equation.6 becaue gen. peed not fed back * M (.0rad / MW ec)(00mw) rad /ec peed ( ) (60) 9.55r. p. m. 3600 ( 9.55) 3590.45r.
More informationReading 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 informationMarch 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 informationMath 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 informationReading 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 informationSOLUTIONS 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 informationMATH 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 informationDimensional Analysis A Tool for Guiding Mathematical Calculations
Dimenional Analyi A Tool for Guiding Mathematical Calculation Dougla A. Kerr Iue 1 February 6, 2010 ABSTRACT AND INTRODUCTION In converting quantitie from one unit to another, we may know the applicable
More information7-5-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 informationSECTION 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(2) Classify the critical points of linear systems and almost linear systems.
Review for Exam 3 Three type of prolem: () Solve the firt order homogeneou linear ytem x Ax () Claify the critical point of linear ytem and almot linear ytem (3) Solve the high order linear equation uing
More informationCHAPTER 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 informationUniform Acceleration Problems Chapter 2: Linear Motion
Name Date Period Uniform Acceleration Problem Chapter 2: Linear Motion INSTRUCTIONS: For thi homework, you will be drawing a coordinate axi (in math lingo: an x-y board ) to olve kinematic (motion) problem.
More informationChapter 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 informationMidterm Review - Part 1
Honor Phyic Fall, 2016 Midterm Review - Part 1 Name: Mr. Leonard Intruction: Complete the following workheet. SHOW ALL OF YOUR WORK. 1. Determine whether each tatement i True or Fale. If the tatement i
More information1 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 informationinto 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 information1. /25 2. /30 3. /25 4. /20 Total /100
Circuit Exam 2 Spring 206. /25 2. /30 3. /25 4. /20 Total /00 Name Pleae write your name at the top of every page! Note: ) If you are tuck on one part of the problem, chooe reaonable value on the following
More informationTMA4125 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 informationME2142/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 informationLecture 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 informationIntroduction 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 informationFeedback Control Systems (FCS)
Feedback Control Sytem (FCS) Lecture19-20 Routh-Herwitz Stability Criterion Dr. Imtiaz Huain email: imtiaz.huain@faculty.muet.edu.pk URL :http://imtiazhuainkalwar.weebly.com/ Stability of Higher Order
More informationME 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 informationME 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 informationChapter 4 Interconnection of LTI Systems
Chapter 4 Interconnection of LTI Sytem 4. INTRODUCTION Block diagram and ignal flow graph are commonly ued to decribe a large feedback control ytem. Each block in the ytem i repreented by a tranfer function,
More informationSIMON 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 informationSection 7.4: Inverse Laplace Transform
Section 74: Inverse Laplace Transform A natural question to ask about any function is whether it has an inverse function We now ask this question about the Laplace transform: given a function F (s), will
More informationChapter 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 informationQuestion 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 informationFactor Analysis with Poisson Output
Factor Analyi with Poion Output Gopal Santhanam Byron Yu Krihna V. Shenoy, Department of Electrical Engineering, Neurocience Program Stanford Univerity Stanford, CA 94305, USA {gopal,byronyu,henoy}@tanford.edu
More informationHomework 7 Solution - AME 30315, Spring s 2 + 2s (s 2 + 2s + 4)(s + 20)
1 Homework 7 Solution - AME 30315, Spring 2015 Problem 1 [10/10 pt] Ue partial fraction expanion to compute x(t) when X 1 () = 4 2 + 2 + 4 Ue partial fraction expanion to compute x(t) when X 2 () = ( )
More informationGiven 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 informationECE382/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 informationSpring 2014 EE 445S Real-Time Digital Signal Processing Laboratory. Homework #0 Solutions on Review of Signals and Systems Material
Spring 4 EE 445S Real-Time Digital Signal Proceing Laboratory Prof. Evan Homework # Solution on Review of Signal and Sytem Material Problem.. Continuou-Time Sinuoidal Generation. In practice, we cannot
More informationNOTE: The items d) and e) of Question 4 gave you bonus marks.
MAE 40 Linear ircuit Summer 2007 Final Solution NOTE: The item d) and e) of Quetion 4 gave you bonu mark. Quetion [Equivalent irciut] [4 mark] Find the equivalent impedance between terminal A and B in
More informationSocial Studies 201 Notes for November 14, 2003
1 Social Studie 201 Note for November 14, 2003 Etimation of a mean, mall ample ize Section 8.4, p. 501. When a reearcher ha only a mall ample ize available, the central limit theorem doe not apply to the
More informationV V The circumflex (^) tells us this is a unit vector
Vector 1 Vector have Direction and Magnitude Mike ailey mjb@c.oregontate.edu Magnitude: V V V V x y z vector.pptx Vector Can lo e Defined a the oitional Difference etween Two oint 3 Unit Vector have a
More informationSIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuits II. Solutions to Assignment 3 February 2005.
SIMON FRASER UNIVERSITY School of Engineering Science ENSC 320 Electric Circuit II Solution to Aignment 3 February 2005. Initial Condition Source 0 V battery witch flip at t 0 find i 3 (t) Component value:
More informationThe Electric Potential Energy
Lecture 6 Chapter 28 Phyic II The Electric Potential Energy Coure webite: http://aculty.uml.edu/andriy_danylov/teaching/phyicii New Idea So ar, we ued vector quantitie: 1. Electric Force (F) Depreed! 2.
More informationName: 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 informationFundamentals of Astrodynamics and Applications 4 th Ed
Fundamental of Atrodynamic and Application 4 th Ed Conolidated Errata July 2, 207 Thi liting i an on-going document of correction and clarification encountered in the book. I appreciate any comment and
More informationThe 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 informationPHYS 110B - HW #6 Spring 2004, Solutions by David Pace Any referenced equations are from Griffiths Problem statements are paraphrased
PHYS B - HW #6 Spring 4, Solution by David Pace Any referenced equation are from Griffith Problem tatement are paraphraed. Problem. from Griffith Show that the following, A µo ɛ o A V + A ρ ɛ o Eq..4 A
More informationCalculation of the temperature of boundary layer beside wall with time-dependent heat transfer coefficient
Ŕ periodica polytechnica Mechanical Engineering 54/1 21 15 2 doi: 1.3311/pp.me.21-1.3 web: http:// www.pp.bme.hu/ me c Periodica Polytechnica 21 RESERCH RTICLE Calculation of the temperature of boundary
More informationFundamentals of Astrodynamics and Applications 4 th Ed
Fundamental of Atrodynamic and Application 4 th Ed Conolidated Errata February 4, 08 Thi liting i an on-going document of correction and clarification encountered in the book. I appreciate any comment
More informationDYNAMICS OF ROTATIONAL MOTION
DYNAMICS OF ROTATIONAL MOTION 10 10.9. IDENTIFY: Apply I. rad/rev SET UP: 0 0. (400 rev/min) 419 rad/ 60 /min EXECUTE: 0 419 rad/ I I (0 kg m ) 11 N m. t 800 EVALUATE: In I, mut be in rad/. 10.. IDENTIFY:
More informationLinear Motion, Speed & Velocity
Add Important Linear Motion, Speed & Velocity Page: 136 Linear Motion, Speed & Velocity NGSS Standard: N/A MA Curriculum Framework (006): 1.1, 1. AP Phyic 1 Learning Objective: 3.A.1.1, 3.A.1.3 Knowledge/Undertanding
More informationName: 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 informationRiemann s Functional Equation is Not Valid and its Implication on the Riemann Hypothesis. Armando M. Evangelista Jr.
Riemann Functional Equation i Not Valid and it Implication on the Riemann Hypothei By Armando M. Evangelita Jr. On November 4, 28 ABSTRACT Riemann functional equation wa formulated by Riemann that uppoedly
More informationIEOR 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 informationSolving Radical Equations
10. Solving Radical Equation Eential Quetion How can you olve an equation that contain quare root? Analyzing a Free-Falling Object MODELING WITH MATHEMATICS To be proficient in math, you need to routinely
More informationChapter 2 Sampling and Quantization. In order to investigate sampling and quantization, the difference between analog
Chapter Sampling and Quantization.1 Analog and Digital Signal In order to invetigate ampling and quantization, the difference between analog and digital ignal mut be undertood. Analog ignal conit of continuou
More informationonline learning Unit Workbook 4 RLC Transients
online learning Pearon BTC Higher National in lectrical and lectronic ngineering (QCF) Unit 5: lectrical & lectronic Principle Unit Workbook 4 in a erie of 4 for thi unit Learning Outcome: RLC Tranient
More informationSingular perturbation theory
Singular perturbation theory Marc R. Rouel June 21, 2004 1 Introduction When we apply the teady-tate approximation (SSA) in chemical kinetic, we typically argue that ome of the intermediate are highly
More informationANSWERS 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 informationEigenvalues and eigenvectors
Eigenvalue and eigenvector Defining and computing uggeted problem olution For each matri give below, find eigenvalue and eigenvector. Give a bai and the dimenion of the eigenpace for each eigenvalue. P:
More informationV V V V. Vectors. Mike Bailey. Vectors have Direction and Magnitude. Magnitude: x y z. Computer Graphics.
1 Vector Mike Bailey mjb@c.oregontate.edu vector.pptx Vector have Direction and Magnitude Magnitude: V V V V x y z 1 Vector Can lo Be Defined a the Poitional Difference Between Two Point 3 ( x, y, z )
More informationChapter 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 informationChapter 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 informationModeling 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 informationBogoliubov 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 informationLinear 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 informationImpulse. calculate the impulse given to an object calculate the change in momentum as the result of an impulse
Add Important Impule Page: 386 Note/Cue Here NGSS Standard: N/A Impule MA Curriculum Framework (2006): 2.5 AP Phyic 1 Learning Objective: 3.D.2.1, 3.D.2.2, 3.D.2.3, 3.D.2.4, 4.B.2.1, 4.B.2.2 Knowledge/Undertanding
More informationExample: Amplifier Distortion
4/6/2011 Example Amplifier Ditortion 1/9 Example: Amplifier Ditortion Recall thi circuit from a previou handout: 15.0 R C =5 K v ( t) = v ( t) o R B =5 K β = 100 _ vi( t ) 58. R E =5 K CUS We found that
More informationThe 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 informationThe machines in the exercise work as follows:
Tik-79.148 Spring 2001 Introduction to Theoretical Computer Science Tutorial 9 Solution to Demontration Exercie 4. Contructing a complex Turing machine can be very laboriou. With the help of machine chema
More informationMain 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 information4e st dt. 0 e st dt. lim. f (t)e st dt. f (t) e st dt + 0. f (t) e. e (2 s)t dt + 0. e (2 s)4 1 ] = 1 = 1. te st dt + = t s e st
Worked Solution Chapter : The Laplace Tranform 6 a F L4] 6 c F L f t] 4 4e t dt e t dt 4 e t 4 ] e t e 4 if > 6 e F L f t] 6 g Uing integration by part, f te t dt f t e t dt + e t dt + e t + 4 4 4 f te
More informationLecture 15 - Current. A Puzzle... Advanced Section: Image Charge for Spheres. Image Charge for a Grounded Spherical Shell
Lecture 15 - Current Puzzle... Suppoe an infinite grounded conducting plane lie at z = 0. charge q i located at a height h above the conducting plane. Show in three different way that the potential below
More informationDIFFERENTIAL EQUATIONS
Matheatic Reviion Guide Introduction to Differential Equation Page of Author: Mark Kudlowki MK HOME TUITION Matheatic Reviion Guide Level: A-Level Year DIFFERENTIAL EQUATIONS Verion : Date: 3-4-3 Matheatic
More informationThe 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 informationPikeville Independent Schools [ALGEBRA 1 CURRICULUM MAP ]
Pikeville Independent School [ALGEBRA 1 CURRICULUM MAP 20162017] Augut X X X 11 12 15 16 17 18 19 22 23 24 25 26 12 37 8 12 29 30 31 13 15 September 1 2 X 6 7 8 9 16 17 18 21 PreAlgebra Review Algebra
More information4e 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 informationA Note on the Sum of Correlated Gamma Random Variables
1 A Note on the Sum of Correlated Gamma Random Variable Joé F Pari Abtract arxiv:11030505v1 [cit] 2 Mar 2011 The um of correlated gamma random variable appear in the analyi of many wirele communication
More informationProperties of Z-transform Transform 1 Linearity a
Midterm 3 (Fall 6 of EEG:. Thi midterm conit of eight ingle-ided page. The firt three page contain variou table followed by FOUR eam quetion and one etra workheet. You can tear out any page but make ure
More informationOnline Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat
Online Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat Thi Online Appendix contain the proof of our reult for the undicounted limit dicued in Section 2 of the paper,
More informationCHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang
CHBE3 ECTURE V APACE TRANSFORM AND TRANSFER FUNCTION Profeor Dae Ryook Yang Spring 8 Dept. of Chemical and Biological Engineering 5- Road Map of the ecture V aplace Tranform and Tranfer function Definition
More information2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim
Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the
More informationThe 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 informationMathematical modeling of control systems. Laith Batarseh. Mathematical modeling of control systems
Chapter two Laith Batareh Mathematical modeling The dynamic of many ytem, whether they are mechanical, electrical, thermal, economic, biological, and o on, may be decribed in term of differential equation
More informationDesign 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 informationMidterm 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 informationRiemann s Functional Equation is Not a Valid Function and Its Implication on the Riemann Hypothesis. Armando M. Evangelista Jr.
Riemann Functional Equation i Not a Valid Function and It Implication on the Riemann Hypothei By Armando M. Evangelita Jr. armando78973@gmail.com On Augut 28, 28 ABSTRACT Riemann functional equation wa
More informationPhysics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014
Phyic 7 Graduate Quantum Mechanic Solution to inal Eam all 0 Each quetion i worth 5 point with point for each part marked eparately Some poibly ueful formula appear at the end of the tet In four dimenion
More informationDigital 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( 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