Power Systems Control Prof. Wonhee Kim. Modeling in the Frequency and Time Domains
|
|
- Andrew Moody
- 5 years ago
- Views:
Transcription
1 Power Systems Control Prof. Wonhee Kim Modeling in the Frequency and Time Domains
2 Laplace Transform Review - Laplace transform - Inverse Laplace transform 2
3 Laplace Transform Review 3
4 Laplace Transform Review 4
5 Laplace Transform Review: - Solution of a Differential Equation Example 2.3) 5
6 System Modeling in Frequency Domain r(t) c(t) r(t) c(t) From input to output: Convolution t * 0 c g r t g r t d 6
7 System Modeling in Frequency Domain Laplace transform r(t) c(t) R(s) C(s) t * 0 c g r t g r t d G s Rs C s Transfer function C s R s G s 7
8 System Modeling in Frequency Domain R(s) C(s) 8
9 System Modeling in Frequency Domain Laplace transform r(t) c(t) R(s) C(s) Example 2.4) dc t 2ct r t dt sc s C s R s 2 Cs 1 Rs s 2 G s 9
10 System Modeling in Frequency Domain Example 2.5) G s C s 1 R s s 2 1 C sg s Rs Rs s 2 1 Rs s 1 1 1/ 2 1/ 2 s s 2 s s 2 G s Rs C s e c t t 10
11 Electrical Network Transfer Functions 11
12 Electrical Network Transfer Functions Example 2.6)
13 Electrical Network Transfer Functions Example 2.6) For the capacitor: For the resistor: For the inductor: Impedance: i t Cdv t / dt I s C scv s
14 Operational Amplifier 14
15 Operational Amplifier 15
16 Operational Amplifier 16
17 Translational Mechanical System Transfer Functions 17
18 Rotational Mechanical System Transfer Functions 18
19 Translational Mechanical System Transfer Functions Example 2.16) 19
20 Nonlinearities 20
21 Nonlinearities - Linearization 21
22 Nonlinearities - Linearization 22
23 Nonlinearities - Linearization 23
24 Some Observation Example Loop equation: I s Vs Ls R 24
25 Some Observation Loop equation: Input State variable I s Vs Ls R State equation: Output 25
26 Some Observation Example Loop equation: Not first order differential equation! 26
27 Some Observation Loop equation: Input State equation: State variable i t, q t Output 27
28 Some Observation Loop equation: Input State equation: State variable v t, v t C R 28
29 Some Observation State variable: or Input State variables must be linearly independent! Linearly dependent: ex) State variable definition v t, i t R v t, v t C i t R, q t 29
30 Some Observation State variable: or Input State variables must be linearly independent! Linearly dependent: ex) State variable definition v t, i t R v t, v t C i t R, q t 30
31 Some Observation 31
32 Some Observation State-space equation 32
33 Some Observation x v v R C Fig. Graphic representation of state space and a state vector 33
34 State-space Equation State-space equation State equation Output equation This representation of a system provides complete knowledge of all variables of the system at any t t 0. 34
35 State-space Equation 2nd order single-input single-output state-space equation: The choice of state variables for a given system is not unique. 35
36 State-space Equation How do we know the minimum number of state variables to select? Typically, the minimum number required equals the order of the differential equation describing the system. State variable i t, q t 36
37 Converting from Transfer Function to State Space Differential equation State variables and state equation (Phase variable form): 37
38 Converting from Transfer Function to State Space Differential equation State-space equation (Phase variable form): Converting is not unique! 38
39 Converting from Transfer Function to State Space Example 2.4) 39
40 Converting from Transfer Function to State Space Example 3.4) 40
41 Converting from State Space to a Transfer Function State-space equation Laplace transform assuming zero initial conditions Transfer Function 41
42 Converting from State Space to a Transfer Function Example 3.6) 42
43 Converting from State Space to a Transfer Function Example 3.6) 43
44 Linear System and Nonlinear System 1) Linear systems: - Linear time-invariant (LTI) system x t Ax t Bu t tc td t y x u - Linear time-varying (LTV) system x t A t x t B t u t tc ttdtt y x u 2) Nonlinear systems: - Nonlinear time-invariant system x t f x t,u t t g tt y x,u - Nonlinear time-varying nonlinear system x t f x t,u t, t t y t g x t,u, t Y s T s C si A 1 B D U s Y s T s C si A 1 B D U s Y s T s C si A 1 B D U s Y s T s C si A 1 B D U s 44
45 Linear System and Nonlinear System 1) Linear systems: - Linear time-invariant (LTI) system x t Ax t Bu t tc td t y x u - Linear time-varying (LTV) system x t A t x t B t u t tc ttdtt y x u 2) Nonlinear systems: - Nonlinear time-invariant system x t f x t,u t t g tt y x,u - Nonlinear time-varying nonlinear system x t f x t,u t, t t y t g x t,u, t x 2 x u y x x 2t x u y x 3 x x u y x 3 t x x e u y x 45
Taking 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 informationI Laplace transform. I Transfer function. I Conversion between systems in time-, frequency-domain, and transfer
EE C128 / ME C134 Feedback Control Systems Lecture Chapter 2 Modeling in the Frequency Domain Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley Lecture
More informationPower System Control
Power System Control Basic Control Engineering Prof. Wonhee Kim School of Energy Systems Engineering, Chung-Ang University 2 Contents Why feedback? System Modeling in Frequency Domain System Modeling in
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 informationSchool of Engineering Faculty of Built Environment, Engineering, Technology & Design
Module Name and Code : ENG60803 Real Time Instrumentation Semester and Year : Semester 5/6, Year 3 Lecture Number/ Week : Lecture 3, Week 3 Learning Outcome (s) : LO5 Module Co-ordinator/Tutor : Dr. Phang
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Systems Prof. Mark Fowler Note Set #9 C-T Systems: Laplace Transform Transfer Function Reading Assignment: Section 6.5 of Kamen and Heck /7 Course Flow Diagram The arrows here show conceptual
More informationLecture 6: Impedance (frequency dependent. resistance in the s- world), Admittance (frequency. dependent conductance in the s- world), and
Lecture 6: Impedance (frequency dependent resistance in the s- world), Admittance (frequency dependent conductance in the s- world), and Consequences Thereof. Professor Ray, what s an impedance? Answers:
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 informationI System variables: states, inputs, outputs, & measurements. I Linear independence. I State space representation
EE C28 / ME C34 Feedback Control Systems Lecture Chapter 3 Modeling in the Time Domain Lecture abstract Alexandre Bayen Department of Electrical Engineering & Computer Science University of California
More informationDr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review
Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics
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 informationChapter 10 AC Analysis Using Phasors
Chapter 10 AC Analysis Using Phasors 10.1 Introduction We would like to use our linear circuit theorems (Nodal analysis, Mesh analysis, Thevenin and Norton equivalent circuits, Superposition, etc.) to
More informationECE2262 Electric Circuit
ECE2262 Electric Circuit Chapter 7: FIRST AND SECOND-ORDER RL AND RC CIRCUITS Response to First-Order RL and RC Circuits Response to Second-Order RL and RC Circuits 1 2 7.1. Introduction 3 4 In dc steady
More informatione st f (t) dt = e st tf(t) dt = L {t f(t)} s
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
More informationLinear System Theory. Wonhee Kim Lecture 1. March 7, 2018
Linear System Theory Wonhee Kim Lecture 1 March 7, 2018 1 / 22 Overview Course Information Prerequisites Course Outline What is Control Engineering? Examples of Control Systems Structure of Control Systems
More informationLinear System Theory
Linear System Theory - Laplace Transform Prof. Robert X. Gao Department of Mechanical Engineering University of Connecticut Storrs, CT 06269 Outline What we ve learned so far: Setting up Modeling Equations
More informationChapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu
Chapter : Time-Domain Representations of Linear Time-Invariant Systems Chih-Wei Liu Outline Characteristics of Systems Described by Differential and Difference Equations Block Diagram Representations State-Variable
More informationElectrical Circuits (2)
Electrical Circuits (2) Lecture 7 Transient Analysis Dr.Eng. Basem ElHalawany Extra Reference for this Lecture Chapter 16 Schaum's Outline Of Theory And Problems Of Electric Circuits https://archive.org/details/theoryandproblemsofelectriccircuits
More informationFundamentals of DC Testing
Fundamentals of DC Testing Aether Lee erigy Japan Abstract n the beginning of this lecture, Ohm s la, hich is the most important electric la regarding DC testing, ill be revieed. Then, in the second section,
More informationState Variable Analysis of Linear Dynamical Systems
Chapter 6 State Variable Analysis of Linear Dynamical Systems 6 Preliminaries In state variable approach, a system is represented completely by a set of differential equations that govern the evolution
More informationLinear System Theory
Linear System Theory Wonhee Kim Lecture 3 Mar. 21, 2017 1 / 38 Overview Recap Nonlinear systems: existence and uniqueness of a solution of differential equations Preliminaries Fields and Vector Spaces
More informationChap 4. State-Space Solutions and
Chap 4. State-Space Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations
More informationLecture 6: Impedance (frequency dependent. resistance in the s-world), Admittance (frequency. dependent conductance in the s-world), and
Lecture 6: Impedance (frequency dependent resistance in the s-world), Admittance (frequency dependent conductance in the s-world), and Consequences Thereof. Professor Ray, what s an impedance? Answers:.
More informationControl Systems. EC / EE / IN. For
Control Systems For EC / EE / IN By www.thegateacademy.com Syllabus Syllabus for Control Systems Basic Control System Components; Block Diagrammatic Description, Reduction of Block Diagrams. Open Loop
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 informationSignals and Systems Chapter 2
Signals and Systems Chapter 2 Continuous-Time Systems Prof. Yasser Mostafa Kadah Overview of Chapter 2 Systems and their classification Linear time-invariant systems System Concept Mathematical transformation
More informationUnit 2: Modeling in the Frequency Domain. Unit 2, Part 4: Modeling Electrical Systems. First Example: Via DE. Resistors, Inductors, and Capacitors
Unit 2: Modeling in the Frequency Domain Part 4: Modeling Electrical Systems Engineering 582: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 20,
More information2.004 Dynamics and Control II Spring 2008
MIT OpenCourseWare http://ocwmitedu 00 Dynamics and Control II Spring 00 For information about citing these materials or our Terms of Use, visit: http://ocwmitedu/terms Massachusetts Institute of Technology
More informationDelhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Serial : CH_EE_B_Network Theory_098 Delhi Noida Bhopal Hyderabad Jaipur Lucknow ndore Pune Bhubaneswar Kolkata Patna Web: E-mail: info@madeeasy.in Ph: 0-56 CLASS TEST 08-9 ELECTCAL ENGNEENG Subject : Network
More informationMEM 255 Introduction to Control Systems: Modeling & analyzing systems
MEM 55 Introduction to Control Systems: Modeling & analyzing systems Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline The Pendulum Micro-machined capacitive accelerometer
More informationProblem Solving 8: Circuits
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics OBJECTIVES Problem Solving 8: Circuits 1. To gain intuition for the behavior of DC circuits with both resistors and capacitors or inductors.
More informationI Poles & zeros. I First-order systems. I Second-order systems. I E ect of additional poles. I E ect of zeros. I E ect of nonlinearities
EE C28 / ME C34 Lecture Chater 4 Time Resonse Alexandre Bayen Deartment of Electrical Engineering & Comuter Science University of California Berkeley Lecture abstract Toics covered in this resentation
More informationCIRCUITS AND ELECTRONICS
6.002x CIRCUITS AND EECTRONICS The Impedance Model!!! Reading: Section 3.3 from text Review Sinusoidal Steady State (SSS) Reading 3., 3.2 v I i cost R C v SSS Focus on sinusoids. Focus on steady state,
More informationControl Systems, Lecture 05
Control Systems, Lecture 05 İbrahim Beklan Küçükdemiral Yıldız Teknik Üniversitesi 2015 1 / 33 Laplace Transform Solution of State Equations In previous sections, systems were modeled in state space, where
More informationBasic. Theory. ircuit. Charles A. Desoer. Ernest S. Kuh. and. McGraw-Hill Book Company
Basic C m ш ircuit Theory Charles A. Desoer and Ernest S. Kuh Department of Electrical Engineering and Computer Sciences University of California, Berkeley McGraw-Hill Book Company New York St. Louis San
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 09-Dec-13 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page
More informationLecture 37: Frequency response. Context
EECS 05 Spring 004, Lecture 37 Lecture 37: Frequency response Prof J. S. Smith EECS 05 Spring 004, Lecture 37 Context We will figure out more of the design parameters for the amplifier we looked at in
More informationPrüfung Regelungstechnik I (Control Systems I) Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 29. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid
More informationModeling and Simulation Revision IV D R. T A R E K A. T U T U N J I P H I L A D E L P H I A U N I V E R S I T Y, J O R D A N
Modeling and Simulation Revision IV D R. T A R E K A. T U T U N J I P H I L A D E L P H I A U N I V E R S I T Y, J O R D A N 2 0 1 7 Modeling Modeling is the process of representing the behavior of a real
More informationNoise - irrelevant data; variability in a quantity that has no meaning or significance. In most cases this is modeled as a random variable.
1.1 Signals and Systems Signals convey information. Systems respond to (or process) information. Engineers desire mathematical models for signals and systems in order to solve design problems efficiently
More informationLaplace Transform in Circuit Analysis
Laplace Transform in Circuit Analysis Laplace Transforms The Laplace transform* is a technique for analyzing linear time-invariant systems such as electrical circuits It provides an alternative functional
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 informationAnalog Signals and Systems and their properties
Analog Signals and Systems and their properties Main Course Objective: Recall course objectives Understand the fundamentals of systems/signals interaction (know how systems can transform or filter signals)
More informationMath 216 Second Midterm 19 March, 2018
Math 26 Second Midterm 9 March, 28 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationErrata to LINEAR CIRCUITS, VOL1 DECARLO/LIN, EDITION 3 CHAPTERS 1 11 (updated 8/23/10)
Errata to LINEAR CIRCUITS, VOL1 DECARLO/LIN, EDITION 3 CHAPTERS 1 11 (updated 8/23/10) page location correction 42 Ch1, P1, statement (e) Figure P.1.3b should be Figure P.1.1b 46 Ch1, P19, statement (c)
More informationModeling of Electrical Elements
Modeling of Electrical Elements Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD This Lecture Contains Modeling of
More informationModeling General Concepts
Modeling General Concepts Basic building blocks of lumped-parameter modeling of real systems mechanical electrical fluid thermal mixed with energy-conversion devices Real devices are modeled as combinations
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response
.. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........
More information7.3 State Space Averaging!
7.3 State Space Averaging! A formal method for deriving the small-signal ac equations of a switching converter! Equivalent to the modeling method of the previous sections! Uses the state-space matrix description
More informationC(s) R(s) 1 C(s) C(s) C(s) = s - T. Ts + 1 = 1 s - 1. s + (1 T) Taking the inverse Laplace transform of Equation (5 2), we obtain
analyses of the step response, ramp response, and impulse response of the second-order systems are presented. Section 5 4 discusses the transient-response analysis of higherorder systems. Section 5 5 gives
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 information信號與系統 Signals and Systems
Spring 2010 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb10 Jun10 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan
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 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 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 informationMATHEMATICAL MODELING OF CONTROL SYSTEMS
1 MATHEMATICAL MODELING OF CONTROL SYSTEMS Sep-14 Dr. Mohammed Morsy Outline Introduction Transfer function and impulse response function Laplace Transform Review Automatic control systems Signal Flow
More informationLecture IV: LTI models of physical systems
BME 171: Signals and Systems Duke University September 5, 2008 This lecture Plan for the lecture: 1 Interconnections of linear systems 2 Differential equation models of LTI systems 3 eview of linear circuit
More informationSection 5 Dynamics and Control of DC-DC Converters
Section 5 Dynamics and ontrol of D-D onverters 5.2. Recap on State-Space Theory x Ax Bu () (2) yxdu u v d ; y v x2 sx () s Ax() s Bu() s ignoring x (0) (3) ( si A) X( s) Bu( s) (4) X s si A BU s () ( )
More information信號與系統 Signals and Systems
Spring 2015 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb15 Jun15 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan
More informationTopic # Feedback Control Systems
Topic #17 16.31 Feedback Control Systems Deterministic LQR Optimal control and the Riccati equation Weight Selection Fall 2007 16.31 17 1 Linear Quadratic Regulator (LQR) Have seen the solutions to the
More informationPHYS225 Lecture 9. Electronic Circuits
PHYS225 Lecture 9 Electronic Circuits Last lecture Field Effect Transistors Voltage controlled resistor Various FET circuits Switch Source follower Current source Similar to BJT Draws no input current
More informationInductance, RL and RLC Circuits
Inductance, RL and RLC Circuits Inductance Temporarily storage of energy by the magnetic field When the switch is closed, the current does not immediately reach its maximum value. Faraday s law of electromagnetic
More informationSystem Modeling. Lecture-2. Emam Fathy Department of Electrical and Control Engineering
System Modeling Lecture-2 Emam Fathy Department of Electrical and Control Engineering email: emfmz@yahoo.com 1 Types of Systems Static System: If a system does not change with time, it is called a static
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 informationChapter 2 Time-Domain Representations of LTI Systems
Chapter 2 Time-Domain Representations of LTI Systems 1 Introduction Impulse responses of LTI systems Linear constant-coefficients differential or difference equations of LTI systems Block diagram representations
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67
1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure
More information1. The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ = Ax + Bu is
ECE 55, Fall 2007 Problem Set #4 Solution The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ Ax + Bu is x(t) e A(t ) x( ) + e A(t τ) Bu(τ)dτ () This formula is extremely important
More informationExact Analysis of a Common-Source MOSFET Amplifier
Exact Analysis of a Common-Source MOSFET Amplifier Consider the common-source MOSFET amplifier driven from signal source v s with Thévenin equivalent resistance R S and a load consisting of a parallel
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 informationControl Systems. Internal Stability - LTI systems. L. Lanari
Control Systems Internal Stability - LTI systems L. Lanari outline LTI systems: definitions conditions South stability criterion equilibrium points Nonlinear systems: equilibrium points examples stable
More informationENGR 2405 Class No Electric Circuits I
ENGR 2405 Class No. 48056 Electric Circuits I Dr. R. Williams Ph.D. rube.williams@hccs.edu Electric Circuit An electric circuit is an interconnec9on of electrical elements Charge Charge is an electrical
More informationChapter 3. Second Order Linear PDEs
Chapter 3. Second Order Linear PDEs 3.1 Introduction The general class of second order linear PDEs are of the form: ax, y)u xx + bx, y)u xy + cx, y)u yy + dx, y)u x + ex, y)u y + f x, y)u = gx, y). 3.1)
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010
[E2.5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 EEE/ISE PART II MEng. BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: 2:00 hours There are FOUR
More informationLast week: analysis of pinion-rack w velocity feedback
Last week: analysis of pinion-rack w velocity feedback Calculation of the steady state error Transfer function: V (s) V ref (s) = 0.362K s +2+0.362K Step input: V ref (s) = s Output: V (s) = s 0.362K s
More informationNetworks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras
Networks and Systems Prof. V. G. K. Murti Department of Electrical Engineering Indian Institute of Technology, Madras Lecture - 34 Network Theorems (1) Superposition Theorem Substitution Theorem The next
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 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 informationMath 216 Final Exam 14 December, 2017
Math 216 Final Exam 14 December, 2017 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More information2.4 Electrical Network Transfer Functions
.4 Electrical Network Transfer Functions 4 Skill-Assessment Exercise.4 PROBLEM: Find the differential equation corresponding to the transfer function, GðsÞ ¼ s þ 1 s þ s þ ANSWER: d c dt þ dc dt þ c ¼
More informationUNIT 1. SIGNALS AND SYSTEM
Page no: 1 UNIT 1. SIGNALS AND SYSTEM INTRODUCTION A SIGNAL is defined as any physical quantity that changes with time, distance, speed, position, pressure, temperature or some other quantity. A SIGNAL
More informationEEE 184: Introduction to feedback systems
EEE 84: Introduction to feedback systems Summary 6 8 8 x 7 7 6 Level() 6 5 4 4 5 5 time(s) 4 6 8 Time (seconds) Fig.. Illustration of BIBO stability: stable system (the input is a unit step) Fig.. step)
More informationAnalog Circuits Prof. Jayanta Mukherjee Department of Electrical Engineering Indian Institute of Technology -Bombay
Analog Circuits Prof. Jayanta Mukherjee Department of Electrical Engineering Indian Institute of Technology -Bombay Week -01 Module -05 Inverting amplifier and Non-inverting amplifier Welcome to another
More informationSwitching Flow Graph Model
Switching Flow Graph Model Keyue Smedley Power Electronics Laboratory University of California Irvine Smedley@uci.edu Switching flow graph model Modeling of single phase active power filter 2 Switching
More informationStability of Parameter Adaptation Algorithms. Big picture
ME5895, UConn, Fall 215 Prof. Xu Chen Big picture For ˆθ (k + 1) = ˆθ (k) + [correction term] we haven t talked about whether ˆθ(k) will converge to the true value θ if k. We haven t even talked about
More information7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system
7 Stability 7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically
More informationEMC Considerations for DC Power Design
EMC Considerations for DC Power Design Tzong-Lin Wu, Ph.D. Department of Electrical Engineering National Sun Yat-sen University Power Bus Noise below 5MHz 1 Power Bus Noise below 5MHz (Solution) Add Bulk
More informationEE100Su08 Lecture #9 (July 16 th 2008)
EE100Su08 Lecture #9 (July 16 th 2008) Outline HW #1s and Midterm #1 returned today Midterm #1 notes HW #1 and Midterm #1 regrade deadline: Wednesday, July 23 rd 2008, 5:00 pm PST. Procedure: HW #1: Bart
More informationChap. 3 Laplace Transforms and Applications
Chap 3 Laplace Transforms and Applications LS 1 Basic Concepts Bilateral Laplace Transform: where is a complex variable Region of Convergence (ROC): The region of s for which the integral converges Transform
More informationINSTRUMENTAL ENGINEERING
INSTRUMENTAL ENGINEERING Subject Code: IN Course Structure Sections/Units Section A Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Section B Section C Section D Section E Section F Section G Section H Section
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signals & Systems Prof. Mark Fowler Note Set #15 C-T Systems: CT Filters & Frequency Response 1/14 Ideal Filters Often we have a scenario where part of the input signal s spectrum comprises what
More informationChap 2. Discrete-Time Signals and Systems
Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,
More informationChapter three. Mathematical Modeling of mechanical end electrical systems. Laith Batarseh
Chapter three Mathematical Modeling of mechanical end electrical systems Laith Batarseh 1 Next Previous Mathematical Modeling of mechanical end electrical systems Dynamic system modeling Definition of
More informationProblem info Geometry model Labelled Objects Results Nonlinear dependencies
Problem info Problem type: Transient Magnetics (integration time: 9.99999993922529E-09 s.) Geometry model class: Plane-Parallel Problem database file names: Problem: circuit.pbm Geometry: Circuit.mod Material
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 informationControl Systems. Frequency domain analysis. L. Lanari
Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic
More informationObservability. It was the property in Lyapunov stability which allowed us to resolve that
Observability We have seen observability twice already It was the property which permitted us to retrieve the initial state from the initial data {u(0),y(0),u(1),y(1),...,u(n 1),y(n 1)} It was the property
More informationLinear dynamical systems with inputs & outputs
EE263 Autumn 215 S. Boyd and S. Lall Linear dynamical systems with inputs & outputs inputs & outputs: interpretations transfer function impulse and step responses examples 1 Inputs & outputs recall continuous-time
More informationMODELING OF CONTROL SYSTEMS
1 MODELING OF CONTROL SYSTEMS Feb-15 Dr. Mohammed Morsy Outline Introduction Differential equations and Linearization of nonlinear mathematical models Transfer function and impulse response function Laplace
More informationSuppose that we have a real phenomenon. The phenomenon produces events (synonym: outcomes ).
p. 5/44 Modeling Suppose that we have a real phenomenon. The phenomenon produces events (synonym: outcomes ). Phenomenon Event, outcome We view a (deterministic) model for the phenomenon as a prescription
More informationSome of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e
Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)
More information