# Modeling and Simulation Revision III 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

Size: px
Start display at page:

Download "Modeling and Simulation Revision III 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"

## Transcription

1 Modeling and Simulation Revision III 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 0 1 4

2 Block Diagrams Block diagram models consist of two fundamental objects: signal blocks and wires. A block is a processing element which operates on input signals and parameters to produce output signals A wire is to transmits a signal from its origination point (usually a block) to its termination point (usually another block). Block diagrams are suitable to represent multidisciplinary models that represent a physical phenomenon. Dr. Tarek A. Tutunji

3 Block Diagram Example [Ref.] Prof. Shetty

4 Block Diagrams Manipulation

5 Block Diagrams Manipulation

6 Block Diagrams: Direct Method Example Consider the transfer function: We can introduce s state variable, x(t), in order to separate the polynomials

7 State Equation The differential equation is: Put the needed integrator blocks: Add the required multipliers to obtain the state equation:

8 Output Equation Repeat the same procedure for the output equation: Connect the two sub-blocks

9 Block Diagram Modeling: Analogy Approach Physical laws are used to predict the behavior (both static and dynamic) of systems. Electrical engineering relies on Ohm s and Kirchoff s laws Mechanical engineering on Newton s law Electromagnetics on Faradays and Lenz s laws Fluids on continuity and Bernoulli s law Based on electrical analogies, we can derive the fundamental equations of systems in five disciplines of engineering: Electrical, Mechanical, Electromagnetic, Fluid, and Thermal. By using this analogy method to first derive the fundamental relationships in a system, the equations then can be represented in block diagram form, allowing secondary and nonlinear effects to be added. This two-step approach is especially useful when modeling large coupled systems using block diagrams.

10 Power and Energy Variables: Effort & Flow [Ref.] Raul Longoria

11 Transitional Mechanical Systems Mechanical movements in a straight line (i.e. linear motion) are called transitional Force displacement Basic Blocks are: Dampers, Masses, and Springs Springs represent the stiffness of the system F kx Dampers (or dashpots) represent the forces opposing to the motion (i.e. friction) F cv velocity Masses represent the inertia F ma mass acceleration Dr. Tarek A. Tutunji

12 Transitional Mechanical Systems Equations for mechanical systems are based on Newton Laws Free body diagram Spring Mass Force Damper ma F - kx - c dx dt Dr. Tarek A. Tutunji

13 Example: Mass-Spring-Damper Note: D is Differentiation 1/D is Integration

14 Example: Two-Mass Mechanical System [Ref.] Prof. Shetty

15 Example: Two-Mass Mechanical System

16 Example: Mechanical Model Consider a two carriage train system x1 x Force Mass 1 k(x1-x) k(x1-x) Mass c v1 m m 1 x x 1 f - k(x k(x x x ) - cx ) - cx 1 c v

17 Example continued Taking the Laplace transform of the equations gives m m 1 s s X X 1 (s) (s) F(s) k(x - 1 k(x (s) - 1 (s) X - X (s)) (s)) - csx - csx Note: Laplace transforms the time domain problem into s-domain (i.e. frequency) L L x(t) X(s) x(t) sx(s) 0 e st x(t)dt (s) 1 (s)

18 Example continued Manipulating the previous two equations, gives the following transfer function (with F as input and V1 as output) V 1 (s) F(s) m 1 m s 3 c m s cs k m m s km km c s kc 1 1 Note: Transfer function is a frequency domain equation that gives the relationship between a specific input to a specific output

19 Example continued Simulation using MATLAB m1= 5; m=0.7; k=0.8; c=0.05; num=[m c k]; den=[m1*m c*m1+c*m k*m1+k*m+c*c *k*c]; sys=tf(num,den); % constructs the transfer function impulse(sys); % plots the impulse response step(sys); % plots the step response bode(sys); % plots the Bode plot

20 Example continued: Impulse response Dr. Tarek A. Tutunji

21 Example continued: Step response Dr. Tarek A. Tutunji

22 Dr. Tarek A. Tutunji Example continued: Bode Plot

23 Example: Motion of Aircraft

24 Rotational Mechanical Systems Consider a mechanical system that involves rotation Torque, T w dq/dt q T Shaft Side View J dw/dt kq cw The torque, T, replaces the force, F The angle, q, replaces the displacement x The angular velocity, w, replaces velocity v The angular acceleration, a, replaces the acceleration a The moment of inertia J, replaces the mass m Dr. Tarek A. Tutunji Top View

25 Rotational Mechanical Systems The mechanics equation becomes spring coefficient angle damper coefficient angular velocity moment of inertia angular acceleration Torque T kθ cω Jα dθ T kθ c J dt d dt θ Dr. Tarek A. Tutunji

26 Example: Rotational-Transitional System Consider a rack-and-pinion system. The rotational motion of the pinion is transformed into transitional motion of the rack w T in pinion r v F rack For simplicity, the spring effects are ignored dω Tin Tout J c1ω dt Dr. Tarek A. Tutunji

27 Example continued The rotational equation is dω Tin Tout J c1ω dt The transitional equation is F cv m dv dt Using the equations T out rf And manipulating the rotational and transitional equations with the input torque, Tin, as inputs and velocity, v, as output, we get ω v/r c T 1 in c r r v J r mr dv dt Dr. Tarek A. Tutunji

28 Example continued Let us take a look at the state space equations In general, where x is the states vector, y is the output vector, and u is the input vector x y Ax Bx Cu Du In our example, we will dω use the states: w and v, dv dt the inputs: T in and F dt the output: v v 0 Manipulating the equations in the previous slide, we get c1 J 0 ω v 1 c 0 ω v m 1 J 0 r J T 1 F m in

29 Conversion: Transitional and Rotational

30 Gear Trains

31 where Gear Trains

32 Electrical Systems: Basic Equations Resistor Ohm s Law Voltage V Ri current Inductor V L di dt Resistance Capacitor Power = Voltage x Current Inductance V i 1 idt C dv C dt Capacitance Dr. Tarek A. Tutunji

33 Kirchoff Laws Equations for electrical systems are based on Kirchoff s Laws 1. Kirchoff current law: Sum of Input currents at node = Sum of output currents. Kirchoff voltage law: Summation of voltage in closed loop equals zero

34 Example: RLC circuit R i L C V V R - + V L - + V c - Using Kirchoff voltage law V Ri L di dt 1 C idt Or V Ri L di dt Vc since i C dv dt c Then V RC dv dt c LC d dt V c V A second order differential equation c

35 RLC MATLAB Code R= ; % R = 1MW L=0.001; % L=1 mh C= ; % C= 1mF num=1; den=[l*c R*C 1]; sys=tf(num,den); bode(sys) Impulse(sys) Step(sys)

36 RLC Simulation: Bode Plot At DC (i.e. frequency = 0), Capacitor is open => Voltage gain is 0dB (i.e. 1 V/V) At high frequency, Capacitor is short => Voltage = 0 Dr. Tarek A. Tutunji

37 RLC Simulation: Impulse Response Input voltage is pulse => Capacitor stores energy And then releases the energy Dr. Tarek A. Tutunji

38 RLC Simulation: Step Response At about.3 seconds, the capacitor Voltage becomes 90% of the 1 Volts Dr. Tarek A. Tutunji

39 Op Amps

40 PM-DC Motor Modeling R i L V in + - Motor T w The electrical equation is V in Ri L di dt V emf where V emf (Back electromagnetic voltage) = k 1 w dω The mechanical equation is T J bω Tload dt where T = k i

41 DC Motor Model: Block Diagram T load V in i T + 1/(Ls+R) + - k - 1/(Js+b) w k 1

42 Simulation Result

43 Fluid Systems Fluid systems can be divided into two categories: Hydraulic: fluid is a liquid and incompressible Pneumatic: fluid is gas and can be compressed The volumetric rate of flow, q, is equivalent to the current The pressure difference, P 1 -P, is equivalent to voltage The basic building blocks for hydraulic systems are: Hydraulic resistance, capacitance, and inertance

44 Hydraulic resistance Hydraulic resistance is the resistance to the fluid flow which occurs as a result of valves or pipe diameter changes The relationship between the volume rate of flow, q, and pressure difference, p 1 -p,is given by Ohm s law p p 1 Rq p 1 p p 1 p pipe valve

45 Hydraulic Capacitance Potential energy stored in a liquid such as height of a liquid in a container q 1 h A p 1 p q Volume Change Volumetric rate of change q V 1 q Ah Cross sectional Area dv dt q 1 q dh A dt height

46 Hydraulic Capacitance p 1 p p hgρ density pressure height gravity Note that p F / A mg / A p Vg / A p hg dh q q A q1 - q dt d p gρ A dt 1 A gρ dp dt By letting the hydraulic capacitance be We get q q 1 C dp dt C A gρ

47 Hydraulic Inertance Equivalent to inductance in electrical systems To accelerate a fluid and increase its velocity a force is required F 1 F p1 p Cross sectional Area F mass 1 = p 1 A Length using F = p A Then p p 1 F1 F ma m ALρ q Av dq I dt dv m dt A Where the Inertance is Lρ I A

48 Hydraulic Example Modeling: an interactive -tank system q in (t) A h 1 (t) A 1 R 1 q h (t) 1 (t) dh1 dt dh dt q (t) q 1 (t) q q h h in 1 1 (t) q (t) q (t) h (t)/r 1 (t) (t) (t) /A /A /R 1 1 R q (t)

49 Hydraulic Example Modeling: Block Diagram Input: q in 1/A 1 - dh1 h h /dt 1 Sum Integrate Sum 1/R 1 1/A Sum Integrate - - q 1 1/A 1 1/A 1/R Dr. Tarek A. Tutunji Output: q

50 Hydraulic Example: Simulation Input, q in, is a step Output, q, is taken to a virtual scope Here, we assume all the Cross sectional areas and the resistances equals 1

51 Hydraulic Example: Simulation

52 Another Form of Analogies Potential and Flow Variables When systems are in motion, the energy can be Increased by an energy-producing source outside the system Redistributed between components within the system Decreased by energy loss through components out of the system. Therefore, a coupled system becomes synonymous with energy transfer between systems. Potential Variable = PV Flow Variable = FV

53 Analogies: FV and PV Flow Variable (FV) Potential Variable (PV) Electrical Current Voltage Mechanical Transitional Force Velocity Mechanical Rotational Torque Angular Velocity Hydraulic Volumetric Flow Rate Pressure Pneumatic Mass Flow Rate Pressure Thermal Heat Flow Rate Temperature

54 Which Analogies to use? Force-Voltage makes more physical sense Graphical Representation: Bond Graphs Force-Current makes mathematical sense Sum of Currents= Zero and Sum of Forces = Zero Graphical Representation: Linear Graphs

55 Conclusion Mathematical Modeling of physical systems is an essential step in the design process Simulation should follow the modeling in order to investigate the system response Mechatronic systems involve different disciplines and therefore an appropriate modeling technique to use is block diagrams Analogies among disciplines can be used to simplify the understanding of different dynamic behaviors Dr. Tarek A. Tutunji

### 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

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

### ECEN 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

### ET3-7: Modelling II(V) Electrical, Mechanical and Thermal Systems

ET3-7: Modelling II(V) Electrical, Mechanical and Thermal Systems Agenda of the Day 1. Resume of lesson I 2. Basic system models. 3. Models of basic electrical system elements 4. Application of Matlab/Simulink

### Modeling of Dynamic Systems: Notes on Bond Graphs Version 1.0 Copyright Diane L. Peters, Ph.D., P.E.

Modeling of Dynamic Systems: Notes on Bond Graphs Version 1.0 Copyright 2015 Diane L. Peters, Ph.D., P.E. Spring 2015 2 Contents 1 Overview of Dynamic Modeling 5 2 Bond Graph Basics 7 2.1 Causality.............................

### AP Physics C Mechanics Objectives

AP Physics C Mechanics Objectives I. KINEMATICS A. Motion in One Dimension 1. The relationships among position, velocity and acceleration a. Given a graph of position vs. time, identify or sketch a graph

### Mechatronics 1: ME 392Q-6 & 348C 31-Aug-07 M.D. Bryant. Analogous Systems. e(t) Se: e. ef = p/i. q = p /I, p = " q C " R p I + e(t)

V + - K R + - - k b V R V L L J + V C M B Analogous Systems i = q. + ω = θ. C -. λ/l = q v = x F T. Se: e e(t) e = p/i R: R 1 I: I e C = q/c C = dq/dt e I = dp/dt Identical dierential equations & bond

### Modeling and Control Overview

Modeling and Control Overview D R. T A R E K A. T U T U N J I A D V A N C E D C O N T R O L S Y S T E M S M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T P H I L A D E L P H I A U N I

### Index. Index. More information. in this web service Cambridge University Press

A-type elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 A-type variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,

### Solved Problems. Electric Circuits & Components. 1-1 Write the KVL equation for the circuit shown.

Solved Problems Electric Circuits & Components 1-1 Write the KVL equation for the circuit shown. 1-2 Write the KCL equation for the principal node shown. 1-2A In the DC circuit given in Fig. 1, find (i)

### Chapter 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

### (Refer Slide Time: 00:01:30 min)

Control Engineering Prof. M. Gopal Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 3 Introduction to Control Problem (Contd.) Well friends, I have been giving you various

### 2.004 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

### Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2)

Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) For all calculations in this book, you can use the MathCad software or any other mathematical software that you are familiar

### ECE2262 Electric Circuits. Chapter 6: Capacitance and Inductance

ECE2262 Electric Circuits Chapter 6: Capacitance and Inductance Capacitors Inductors Capacitor and Inductor Combinations Op-Amp Integrator and Op-Amp Differentiator 1 CAPACITANCE AND INDUCTANCE Introduces

### Inductance, 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

### Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits

Inductance, Inductors, RL Circuits & RC Circuits, LC, and RLC Circuits Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the timevarying

### Linear 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

### MATHEMATICAL MODELING OF DYNAMIC SYSTEMS

MTHEMTIL MODELIN OF DYNMI SYSTEMS Mechanical Translational System 1. Spring x(t) k F S (t) k x(t) x i (t) k x o (t) 2. Damper x(t) x i (t) x o (t) c c 3. Mass x(t) F(t) m EXMPLE I Produce the block diagram

### Solution for Fq. A. up B. down C. east D. west E. south

Solution for Fq A proton traveling due north enters a region that contains both a magnetic field and an electric field. The electric field lines point due west. It is observed that the proton continues

### ENGG4420 LECTURE 7. CHAPTER 1 BY RADU MURESAN Page 1. September :29 PM

CHAPTER 1 BY RADU MURESAN Page 1 ENGG4420 LECTURE 7 September 21 10 2:29 PM MODELS OF ELECTRIC CIRCUITS Electric circuits contain sources of electric voltage and current and other electronic elements such

### ECE2262 Electric Circuits. Chapter 6: Capacitance and Inductance

ECE2262 Electric Circuits Chapter 6: Capacitance and Inductance Capacitors Inductors Capacitor and Inductor Combinations 1 CAPACITANCE AND INDUCTANCE Introduces two passive, energy storing devices: Capacitors

### Springs and Dampers. MCE371: Vibrations. Prof. Richter. Department of Mechanical Engineering. Handout 2 Fall 2017

MCE371: Vibrations Prof. Richter Department of Mechanical Engineering Handout 2 Fall 2017 Spring Law : One End Fixed Ideal linear spring law: f = kx. What are the units of k? More generally: f = F(x) nonlinear

### Physics for Scientists & Engineers 2

Electromagnetic Oscillations Physics for Scientists & Engineers Spring Semester 005 Lecture 8! We have been working with circuits that have a constant current a current that increases to a constant current

### Here are some internet links to instructional and necessary background materials:

The general areas covered by the University Physics course are subdivided into major categories. For each category, answer the conceptual questions in the form of a short paragraph. Although fewer topics

### Scanned by CamScanner

Scanned by CamScanner Scanned by CamScanner t W I w v 6.00-fall 017 Midterm 1 Name Problem 3 (15 pts). F the circuit below, assume that all equivalent parameters are to be found to the left of port

### Introduction to AC Circuits (Capacitors and Inductors)

Introduction to AC Circuits (Capacitors and Inductors) Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/

### Introduction to Controls

EE 474 Review Exam 1 Name Answer each of the questions. Show your work. Note were essay-type answers are requested. Answer with complete sentences. Incomplete sentences will count heavily against the grade.

### Noise - 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

### Electrical Machine & Automatic Control (EEE-409) (ME-II Yr) UNIT-3 Content: Signals u(t) = 1 when t 0 = 0 when t <0

Electrical Machine & Automatic Control (EEE-409) (ME-II Yr) UNIT-3 Content: Modeling of Mechanical : linear mechanical elements, force-voltage and force current analogy, and electrical analog of simple

### Ch. 23 Electromagnetic Induction, AC Circuits, And Electrical Technologies

Ch. 23 Electromagnetic Induction, AC Circuits, And Electrical Technologies Induced emf - Faraday s Experiment When a magnet moves toward a loop of wire, the ammeter shows the presence of a current When

### Chapter 32. Inductance

Chapter 32 Inductance Joseph Henry 1797 1878 American physicist First director of the Smithsonian Improved design of electromagnet Constructed one of the first motors Discovered self-inductance Unit of

### Review: control, feedback, etc. Today s topic: state-space models of systems; linearization

Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered

### Exam 3 Solutions. The induced EMF (magnitude) is given by Faraday s Law d dt dt The current is given by

PHY049 Spring 008 Prof. Darin Acosta Prof. Selman Hershfield April 9, 008. A metal rod is forced to move with constant velocity of 60 cm/s [or 90 cm/s] along two parallel metal rails, which are connected

### Chapter 8. Model of the Accelerometer. 8.1 The static model 8.2 The dynamic model 8.3 Sensor System simulation

Chapter 8. Model of the Accelerometer 8.1 The static model 8.2 The dynamic model 8.3 Sensor System simulation 8.3 Sensor System Simulation In order to predict the behavior of the mechanical sensor in combination

### Electromagnetic Induction (Chapters 31-32)

Electromagnetic Induction (Chapters 31-3) The laws of emf induction: Faraday s and Lenz s laws Inductance Mutual inductance M Self inductance L. Inductors Magnetic field energy Simple inductive circuits

### Contents. Dynamics and control of mechanical systems. Focus on

Dynamics and control of mechanical systems Date Day 1 (01/08) Day 2 (03/08) Day 3 (05/08) Day 4 (07/08) Day 5 (09/08) Day 6 (11/08) Content Review of the basics of mechanics. Kinematics of rigid bodies

### The basic principle to be used in mechanical systems to derive a mathematical model is Newton s law,

Chapter. DYNAMIC MODELING Understanding the nature of the process to be controlled is a central issue for a control engineer. Thus the engineer must construct a model of the process with whatever information

### SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER 1 EXAMINATIONS 2012/2013 XE121. ENGINEERING CONCEPTS (Test)

s SCHOOL OF COMPUTING, ENGINEERING AND MATHEMATICS SEMESTER EXAMINATIONS 202/203 XE2 ENGINEERING CONCEPTS (Test) Time allowed: TWO hours Answer: Attempt FOUR questions only, a maximum of TWO questions

### Basic Electronics. Introductory Lecture Course for. Technology and Instrumentation in Particle Physics Chicago, Illinois June 9-14, 2011

Basic Electronics Introductory Lecture Course for Technology and Instrumentation in Particle Physics 2011 Chicago, Illinois June 9-14, 2011 Presented By Gary Drake Argonne National Laboratory drake@anl.gov

### SUBJECT & PEDAGOGICAL CONTENT STANDARDS FOR PHYSICS TEACHERS (GRADES 9-10)

SUBJECT & PEDAGOGICAL CONTENT STANDARDS FOR PHYSICS TEACHERS (GRADES 9-10) JULY 2014 2 P a g e 1) Standard 1: Content Knowledge for Grade 9-10 Physics Teacher Understands Models and Scales G9-10PS1.E1.1)

### Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current.

Inductance Self-inductance A time-varying current in a circuit produces an induced emf opposing the emf that initially set up the time-varying current. Basis of the electrical circuit element called an

### Model of a DC Generator Driving a DC Motor (which propels a car)

Model of a DC Generator Driving a DC Motor (which propels a car) John Hung 5 July 2011 The dc is connected to the dc as illustrated in Fig. 1. Both machines are of permanent magnet type, so their respective

### Alternating Current. Symbol for A.C. source. A.C.

Alternating Current Kirchoff s rules for loops and junctions may be used to analyze complicated circuits such as the one below, powered by an alternating current (A.C.) source. But the analysis can quickly

### ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT

Chapter 31: ELECTROMAGNETIC OSCILLATIONS AND ALTERNATING CURRENT 1 A charged capacitor and an inductor are connected in series At time t = 0 the current is zero, but the capacitor is charged If T is the

### Oscillations. Tacoma Narrow Bridge: Example of Torsional Oscillation

Oscillations Mechanical Mass-spring system nd order differential eq. Energy tossing between mass (kinetic energy) and spring (potential energy) Effect of friction, critical damping (shock absorber) Simple

### Louisiana State University Physics 2102, Exam 3 April 2nd, 2009.

PRINT Your Name: Instructor: Louisiana State University Physics 2102, Exam 3 April 2nd, 2009. Please be sure to PRINT your name and class instructor above. The test consists of 4 questions (multiple choice),

### INF5490 RF MEMS. LN03: Modeling, design and analysis. Spring 2008, Oddvar Søråsen Department of Informatics, UoO

INF5490 RF MEMS LN03: Modeling, design and analysis Spring 2008, Oddvar Søråsen Department of Informatics, UoO 1 Today s lecture MEMS functional operation Transducer principles Sensor principles Methods

### General Response of Second Order System

General Response of Second Order System Slide 1 Learning Objectives Learn to analyze a general second order system and to obtain the general solution Identify the over-damped, under-damped, and critically

### FEEDBACK CONTROL SYSTEMS

FEEDBAC CONTROL SYSTEMS. Control System Design. Open and Closed-Loop Control Systems 3. Why Closed-Loop Control? 4. Case Study --- Speed Control of a DC Motor 5. Steady-State Errors in Unity Feedback Control

### Automatic Control Systems. -Lecture Note 15-

-Lecture Note 15- Modeling of Physical Systems 5 1/52 AC Motors AC Motors Classification i) Induction Motor (Asynchronous Motor) ii) Synchronous Motor 2/52 Advantages of AC Motors i) Cost-effective ii)

### Mechatronics Engineering. Li Wen

Mechatronics Engineering Li Wen Bio-inspired robot-dc motor drive Unstable system Mirko Kovac,EPFL Modeling and simulation of the control system Problems 1. Why we establish mathematical model of the control

### Lecture 1. Electrical Transport

Lecture 1. Electrical Transport 1.1 Introduction * Objectives * Requirements & Grading Policy * Other information 1.2 Basic Circuit Concepts * Electrical l quantities current, voltage & power, sign conventions

### System 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

### r where the electric constant

1.0 ELECTROSTATICS At the end of this topic, students will be able to: 10 1.1 Coulomb s law a) Explain the concepts of electrons, protons, charged objects, charged up, gaining charge, losing charge, charging

### ELECTRO MAGNETIC INDUCTION

ELECTRO MAGNETIC INDUCTION 1) A Circular coil is placed near a current carrying conductor. The induced current is anti clock wise when the coil is, 1. Stationary 2. Moved away from the conductor 3. Moved

### School 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

### 2.5 Translational Mechanical System Transfer Functions 61. FIGURE 2.14 Electric circuit for Skill- Assessment Exercise 2.6

.5 Translational echanical System Transfer Functions 1 1Ω 1H 1Ω + v(t) + 1 H 1 H v L (t) FIGURE.14 Electric circuit for Skill- Assessment Exercise. ANSWER: V L ðsþ=vðsþ ¼ðs þ s þ 1Þ=ðs þ 5s þ Þ The complete

### Physics Will Farmer. May 5, Physics 1120 Contents 2

Physics 1120 Will Farmer May 5, 2013 Contents Physics 1120 Contents 2 1 Charges 3 1.1 Terms................................................... 3 1.2 Electric Charge..............................................

### Assessment Schedule 2015 Physics: Demonstrate understanding of electrical systems (91526)

NCEA Level 3 Physics (91526) 2015 page 1 of 6 Assessment Schedule 2015 Physics: Demonstrate understanding of electrical systems (91526) Evidence Q Evidence Achievement Achievement with Merit Achievement

### MATH 312 Section 3.1: Linear Models

MATH 312 Section 3.1: Linear Models Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Outline 1 Population Growth 2 Newton s Law of Cooling 3 Kepler s Law Second Law of Planetary Motion 4

### 2.004 Dynamics and Control II Spring 2008

MT OpenCourseWare http://ocwmitedu 200 Dynamics and Control Spring 200 For information about citing these materials or our Terms of Use, visit: http://ocwmitedu/terms Massachusetts nstitute of Technology

### Mixing Problems. Solution of concentration c 1 grams/liter flows in at a rate of r 1 liters/minute. Figure 1.7.1: A mixing problem.

page 57 1.7 Modeling Problems Using First-Order Linear Differential Equations 57 For Problems 33 38, use a differential equation solver to determine the solution to each of the initial-value problems and

### Energy Storage Elements: Capacitors and Inductors

CHAPTER 6 Energy Storage Elements: Capacitors and Inductors To this point in our study of electronic circuits, time has not been important. The analysis and designs we have performed so far have been static,

### Version 001 CIRCUITS holland (1290) 1

Version CIRCUITS holland (9) This print-out should have questions Multiple-choice questions may continue on the next column or page find all choices before answering AP M 99 MC points The power dissipated

### AC Circuits Homework Set

Problem 1. In an oscillating LC circuit in which C=4.0 μf, the maximum potential difference across the capacitor during the oscillations is 1.50 V and the maximum current through the inductor is 50.0 ma.

### SUMMARY Phys 2523 (University Physics II) Compiled by Prof. Erickson. F e (r )=q E(r ) dq r 2 ˆr = k e E = V. V (r )=k e r = k q i. r i r.

SUMMARY Phys 53 (University Physics II) Compiled by Prof. Erickson q 1 q Coulomb s Law: F 1 = k e r ˆr where k e = 1 4π =8.9875 10 9 N m /C, and =8.85 10 1 C /(N m )isthepermittivity of free space. Generally,

### DcMotor_ Model Help File

Name of Model: DcMotor_021708 Author: Vladimir L. Chervyakov Date: 2002-10-26 Executable file name DcMotor_021708.vtm Version number: 1.0 Description This model represents a Nonlinear model of a permanent

### Chapter 31 Electromagnetic Oscillations and Alternating Current LC Oscillations, Qualitatively

Chapter 3 Electromagnetic Oscillations and Alternating Current LC Oscillations, Qualitatively In the LC circuit the charge, current, and potential difference vary sinusoidally (with period T and angular

### CHAPTER FOUR MUTUAL INDUCTANCE

CHAPTER FOUR MUTUAL INDUCTANCE 4.1 Inductance 4.2 Capacitance 4.3 Serial-Parallel Combination 4.4 Mutual Inductance 4.1 Inductance Inductance (L in Henry is the circuit parameter used to describe an inductor.

### AC vs. DC Circuits. Constant voltage circuits. The voltage from an outlet is alternating voltage

Circuits AC vs. DC Circuits Constant voltage circuits Typically referred to as direct current or DC Computers, logic circuits, and battery operated devices are examples of DC circuits The voltage from

### Slide 1 / 26. Inductance by Bryan Pflueger

Slide 1 / 26 Inductance 2011 by Bryan Pflueger Slide 2 / 26 Mutual Inductance If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one

### Lecture Fluid system elements

Lecture 8.1 Fluid system elements volumetric flowrate pressure drop Detailed distributed models of fluids, such as the Navier-Stokes equations, are necessary for understanding many aspects of fluid systems

### Module 25: Outline Resonance & Resonance Driven & LRC Circuits Circuits 2

Module 25: Driven RLC Circuits 1 Module 25: Outline Resonance & Driven LRC Circuits 2 Driven Oscillations: Resonance 3 Mass on a Spring: Simple Harmonic Motion A Second Look 4 Mass on a Spring (1) (2)

### Part 4: Electromagnetism. 4.1: Induction. A. Faraday's Law. The magnetic flux through a loop of wire is

1 Part 4: Electromagnetism 4.1: Induction A. Faraday's Law The magnetic flux through a loop of wire is Φ = BA cos θ B A B = magnetic field penetrating loop [T] A = area of loop [m 2 ] = angle between field

### 1 2 U CV. K dq I dt J nqv d J V IR P VI

o 5 o T C T F 3 9 T K T o C 73.5 L L T V VT Q mct nct Q F V ml F V dq A H k TH TC L pv nrt 3 Ktr nrt 3 CV R ideal monatomic gas 5 CV R ideal diatomic gas w/o vibration V W pdv V U Q W W Q e Q Q e Carnot

### EDEXCEL NATIONALS UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES. ASSIGNMENT No. 3 - ELECTRO MAGNETIC INDUCTION

EDEXCEL NATIONALS UNIT 5 - ELECTRICAL AND ELECTRONIC PRINCIPLES ASSIGNMENT No. 3 - ELECTRO MAGNETIC INDUCTION NAME: I agree to the assessment as contained in this assignment. I confirm that the work submitted

### ECE2262 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

### EM Oscillations. David J. Starling Penn State Hazleton PHYS 212

I ve got an oscillating fan at my house. The fan goes back and forth. It looks like the fan is saying No. So I like to ask it questions that a fan would say no to. Do you keep my hair in place? Do you

### Rotational Systems, Gears, and DC Servo Motors

Rotational Systems Rotational Systems, Gears, and DC Servo Motors Rotational systems behave exactly like translational systems, except that The state (angle) is denoted with rather than x (position) Inertia

### EE102 Homework 2, 3, and 4 Solutions

EE12 Prof. S. Boyd EE12 Homework 2, 3, and 4 Solutions 7. Some convolution systems. Consider a convolution system, y(t) = + u(t τ)h(τ) dτ, where h is a function called the kernel or impulse response of

### Exercise 5 - Hydraulic Turbines and Electromagnetic Systems

Exercise 5 - Hydraulic Turbines and Electromagnetic Systems 5.1 Hydraulic Turbines Whole courses are dedicated to the analysis of gas turbines. For the aim of modeling hydraulic systems, we analyze here

### PHYS General Physics for Engineering II FIRST MIDTERM

Çankaya University Department of Mathematics and Computer Sciences 2010-2011 Spring Semester PHYS 112 - General Physics for Engineering II FIRST MIDTERM 1) Two fixed particles of charges q 1 = 1.0µC and

### Modeling 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

### 2. The following diagram illustrates that voltage represents what physical dimension?

BioE 1310 - Exam 1 2/20/2018 Answer Sheet - Correct answer is A for all questions 1. A particular voltage divider with 10 V across it consists of two resistors in series. One resistor is 7 KΩ and the other

### EE292: Fundamentals of ECE

EE292: Fundamentals of ECE Fall 2012 TTh 10:00-11:15 SEB 1242 Lecture 14 121011 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review Steady-State Analysis RC Circuits RL Circuits 3 DC Steady-State

### MEM 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

### Physics For Scientists and Engineers A Strategic Approach 3 rd Edition, AP Edition, 2013 Knight

For Scientists and Engineers A Strategic Approach 3 rd Edition, AP Edition, 2013 Knight To the Advanced Placement Topics for C *Advanced Placement, Advanced Placement Program, AP, and Pre-AP are registered

### 2002 Prentice Hall, Inc. Gene F. Franklin, J. David Powell, Abbas Emami-Naeini Feedback Control of Dynamic Systems, 4e

u Figure 2.1 Cruise-control model x Friction force bx m x u Figure 2.2 Free-body diagram for cruise control S P 278 Figure 2.3 Automobile suspension y m 2 k s b v car x m 1 k w Road surface r Inertial

### : TEACHING DIFFERENTIAL EQUATIONS WITH AN ENGINEERING FOCUS

2006-915: TEACHING DIFFERENTIAL EQUATIONS WITH AN ENGINEERING FOCUS Stephen Pennell, University of Massachusetts-Lowell Stephen Pennell is a Professor in the Department of Mathematical Sciences at the

### Exam 2 Solutions. Note that there are several variations of some problems, indicated by choices in parentheses.

Exam 2 Solutions Note that there are several variations of some problems, indicated by choices in parentheses. Problem 1 Part of a long, straight insulated wire carrying current i is bent into a circular

### RLC Circuit (3) We can then write the differential equation for charge on the capacitor. The solution of this differential equation is

RLC Circuit (3) We can then write the differential equation for charge on the capacitor The solution of this differential equation is (damped harmonic oscillation!), where 25 RLC Circuit (4) If we charge

### Some 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)

### Figure Circuit for Question 1. Figure Circuit for Question 2

Exercises 10.7 Exercises Multiple Choice 1. For the circuit of Figure 10.44 the time constant is A. 0.5 ms 71.43 µs 2, 000 s D. 0.2 ms 4 Ω 2 Ω 12 Ω 1 mh 12u 0 () t V Figure 10.44. Circuit for Question

### b) (4) How large is the current through the 2.00 Ω resistor, and in which direction?

General Physics II Exam 2 - Chs. 19 21 - Circuits, Magnetism, EM Induction - Sep. 29, 2016 Name Rec. Instr. Rec. Time For full credit, make your work clear. Show formulas used, essential steps, and results

### 2.4 Harmonic Oscillator Models

2.4 Harmonic Oscillator Models In this section we give three important examples from physics of harmonic oscillator models. Such models are ubiquitous in physics, but are also used in chemistry, biology,

### Module 24: Outline. Expt. 8: Part 2:Undriven RLC Circuits

Module 24: Undriven RLC Circuits 1 Module 24: Outline Undriven RLC Circuits Expt. 8: Part 2:Undriven RLC Circuits 2 Circuits that Oscillate (LRC) 3 Mass on a Spring: Simple Harmonic Motion (Demonstration)