# Definitions. Decade: A ten-to-one range of frequency. On a log scale, each 10X change in frequency requires the same distance on the scale.

Save this PDF as:

Size: px
Start display at page:

Download "Definitions. Decade: A ten-to-one range of frequency. On a log scale, each 10X change in frequency requires the same distance on the scale."

## Transcription

1 Circuits II EECS 3220 Lecture notes on making Bode plots Definitions Network Transfer Function: The function H s Xout s X in s where X out represents the voltage or current response of the network to X in, which is a voltage or current stimulus to the network. The frequency variable s is complex frequency (in radians/s) if Laplace analysis is used, or it is frequency j (in radians/s) if phasor analysis is used. This transfer function is expressed as the quotient of numerator and denominator polynomials in s. Frequency Response Plot: Graphs of 20 log 10 H j and H j versus frequency. These are typically placed on linear scales when the range of frequency is narrow, or semi-log scales when the range of frequency is large. The frequency response plot is based on a steady-state sinusoid and may be measured or calculated data. Bode Plot: An approximation to the exact frequency response plot made using only semi-log graph paper and a straight edge. The magnitude portion of a Bode plot consists only of straight lines having slopes of k 20 db/decade, where k is an integer which may be negative, zero, or positive. The angle portion of the Bode plot consists only of straight lines having slopes of k 45 degrees/decade, where again k is an integer. Decade: A ten-to-one range of frequency. On a log scale, each 10X change in frequency requires the same distance on the scale. Octave: A two-to-one range of frequency. (In music, eight notes are spaced out over an octave. If the half-steps are also included, there are twelve steps in an octave, evenly spaced on a logarithmic scale in frequency.) Within a small round-off error, the slope 20 db/decade is equal to 6 db/octave. Zero: A root of the numerator s-polynomial of the transfer function. It is typical to find zeros with either a positive or a negative real part. Pole: A root of the denominator s-polynomial of the transfer function. It is typical to find poles with a negative real part. A pole having a positive real part indicates an unstable system, which is unlikely to be practical. Singularities: A collective term for the poles and zeros of a transfer function. 1 of 6

2 Network Analysis A linear circuit having L, R, C, coupled inductors and linearly-dependent sources may be analyzed for its transfer function by replacing each element with its frequency-domain equivalent impedance (R, sl, 1/sC, etc.). The outcome invariably can be expressed in the form: H s Xout s X in s K hf s m as m 1 z s n bs n 1 p wherem n (1) By factoring (1) to determine its poles and zeros, (1) can be rewritten in the following more-insightful form: H s Xout s X in s K hf (s z 1 )(s z 2 ) (s z m ) (s p 1 )(s p 2 ) (s p n ) (2) When this form is obtained, a great deal can be known about the network behavior and its frequency response function by merely stating the list of its singularities (poles and zeros). Equation (2) assumes that all of the singularities are purely real. Many circuits will produce a pair of poles or zeros which are complex conjugates. In these cases, it is more helpful to just keep the conjugate pair together as the second-order term: s 2 bs c, where b 2 < 4c. This discussion will consider only real-valued poles and zeros. A quick check on your network function (1) or (2) can be made by looking back at the circuit itself. The constant "K" in these expressions is the high-frequency limit of the transfer function, which can often be verified by directly examining the circuit itself. (At high frequency, inductors become open circuits, and capacitors short circuits.) Be aware that these functions can be reorganized to display their low-frequency limits instead, such as the following: H s Xout s 1 z s X in s K 1 1 z s 2 (1 s lf zm ) 1 s p 1 1 s p 2 1 s pn (3) Another quick check on validity of your transfer function is to count the number of independent energy-storing elements (capacitors and inductors) in your circuit. This number is equal to n, the number of poles. Note that two capacitors in parallel are not independent, and only add one pole. There are also obscure ways for non-independance; for example, three capacitors in a loop will add two to the number of circuit poles, not three. 2 of 6

4 If a single valid point on the magnitude graph can be located, and the slope of the plot as it passes through that point is known, the entire plot can be filled in by starting from this point and drawing straight lines, adjusting their slopes up and down as (the frequency magnitudes of) the zeros and poles are passed. Note that zeros of either sign cause an upturn, and poles cause a downturn in slope. In the case of (4), the low-frequency magnitude can be found by taking the limit for small s (low frequency). lim H s 100 s so H j s (5) Therefore, the magnitude plot of Example 1 may be started with the point ( = 0.1, +20 db). Equation (5) also indicates that the magnitude is zero at zero frequency, but this location cannot be found on a Bode plot. The best way to think of a single zero at 0 frequency is as an upturn in slope of 20 db/dec far off the left side of the graph. The far left side of the plot then appears as a rising line with this slope, passing through the point (0.1, +20 db), and changing slope at the frequency corresponding to each singularity on the list as it is encountered. Note that the zero corresponding to 1000 rad/s cancels the effect of the pole corresponding to that same frequency. The slope at the far right side of the graph should be -20 db/dec. This could be thought of as the effect of the single zero at "infinity," a location off the far right side of the graph. The initial slope on the far left side of plot will be +p 20 db/dec, where p is the number zeros at 0. Likewise, the ultimate slope at the far right side of the plot will -q 20 db/dec, where q is the number of zeros at "infinity" needed to make the list of zeros equal in length to the list of poles. 4 of 6

5 The phase graph is plotted in a similar manner. In this case, what is being plotted is the summation of phase terms from the numerator of H, minus phase terms from the denominator of H. The form is the following: H j arctan( z 1 ) arctan( z 2 ) arctan( p 1 ) arctan( p 2 ) (6) Note that the negative real zeros will contribute between 0 and 90 degrees to the result, positive real zeros will contribute 0 to -90 degrees, and negative real poles will contribute 0 to -90 degrees. The asymptotic behavior for a negative zero is 0 degrees up to z 10, then an upward slope of +45 degrees/decade to z 10, then the final contribution of +90 degrees. The contribution of a negative zero is +45 degrees at the frequency corresponding to the zero. The asymptotic behavior for a positive zero is 0 degrees up to, then an downward slope of -45 degrees/decade to z 10, then the final contribution of -90 degrees. The contribution of a positive zero is -45 degrees at the frequency corresponding to the zero. The asymptotic behavior for a negative pole is 0 degrees up to p 10, then a downward slope of -45 degrees/decade to p 10, then the final contribution of -90 degrees. The contribution of a negative pole is -45 degrees at the frequency corresponding to the pole. The composite phase curve can be obtained by finding one correct starting point, and then extending the graph in both directions using the slope changes given above. In this case, slopes change at one-tenth and ten-times the frequency corresponding to the singularity, so in many practical systems several singularities may be contributing to the net result at a given frequency. The Bode plot of magnitude and phase of the transfer function data given in Example 1 is shown on page 6. z 10 5 of 6

6 6 of 6

### Steady State Frequency Response Using Bode Plots

School of Engineering Department of Electrical and Computer Engineering 332:224 Principles of Electrical Engineering II Laboratory Experiment 3 Steady State Frequency Response Using Bode Plots 1 Introduction

### Dynamic circuits: Frequency domain analysis

Electronic Circuits 1 Dynamic circuits: Contents Free oscillation and natural frequency Transfer functions Frequency response Bode plots 1 System behaviour: overview 2 System behaviour : review solution

### Frequency Response Analysis

Frequency Response Analysis Consider let the input be in the form Assume that the system is stable and the steady state response of the system to a sinusoidal inputdoes not depend on the initial conditions

### Review of Linear Time-Invariant Network Analysis

D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x

### 16.30/31, Fall 2010 Recitation # 2

16.30/31, Fall 2010 Recitation # 2 September 22, 2010 In this recitation, we will consider two problems from Chapter 8 of the Van de Vegte book. R + - E G c (s) G(s) C Figure 1: The standard block diagram

### Laplace Transform Analysis of Signals and Systems

Laplace Transform Analysis of Signals and Systems Transfer Functions Transfer functions of CT systems can be found from analysis of Differential Equations Block Diagrams Circuit Diagrams 5/10/04 M. J.

### ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques

CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques [] For the following system, Design a compensator such

### ESE319 Introduction to Microelectronics. Feedback Basics

Feedback Basics Stability Feedback concept Feedback in emitter follower One-pole feedback and root locus Frequency dependent feedback and root locus Gain and phase margins Conditions for closed loop stability

### UNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences. EE105 Lab Experiments

UNIVERSITY OF CALIFORNIA AT BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences EE15 Lab Experiments Bode Plot Tutorial Contents 1 Introduction 1 2 Bode Plots Basics

### Transient response via gain adjustment. Consider a unity feedback system, where G(s) = 2. The closed loop transfer function is. s 2 + 2ζωs + ω 2 n

Design via frequency response Transient response via gain adjustment Consider a unity feedback system, where G(s) = ωn 2. The closed loop transfer function is s(s+2ζω n ) T(s) = ω 2 n s 2 + 2ζωs + ω 2

### Asymptote. 2 Problems 2 Methods

Asymptote Problems Methods Problems Assume we have the ollowing transer unction which has a zero at =, a pole at = and a pole at =. We are going to look at two problems: problem is where >> and problem

### Learn2Control Laboratory

Learn2Control Laboratory Version 3.2 Summer Term 2014 1 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should

### Dr 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

### Network Graphs and Tellegen s Theorem

Networ Graphs and Tellegen s Theorem The concepts of a graph Cut sets and Kirchhoff s current laws Loops and Kirchhoff s voltage laws Tellegen s Theorem The concepts of a graph The analysis of a complex

### Frequency domain analysis

Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011

### A positive value is obtained, so the current is counterclockwise around the circuit.

Chapter 7. (a) Let i be the current in the circuit and take it to be positive if it is to the left in. We use Kirchhoff s loop rule: ε i i ε 0. We solve for i: i ε ε + 6. 0 050.. 4.0Ω+ 80. Ω positive value

### The RC Time Constant

The RC Time Constant Objectives When a direct-current source of emf is suddenly placed in series with a capacitor and a resistor, there is current in the circuit for whatever time it takes to fully charge

### Lecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore

Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.

### ESE319 Introduction to Microelectronics Bode Plot Review High Frequency BJT Model

Bode Plot Review High Frequency BJT Model 1 Logarithmic Frequency Response Plots (Bode Plots) Generic form of frequency response rational polynomial, where we substitute jω for s: H s=k sm a m 1 s m 1

### Homework 7 - Solutions

Homework 7 - Solutions Note: This homework is worth a total of 48 points. 1. Compensators (9 points) For a unity feedback system given below, with G(s) = K s(s + 5)(s + 11) do the following: (c) Find the

### 8 sin 3 V. For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0.

For the circuit given, determine the voltage v for all time t. Assume that no energy is stored in the circuit before t = 0. Spring 2015, Exam #5, Problem #1 4t Answer: e tut 8 sin 3 V 1 For the circuit

### Section 3.4 Rational Functions

88 Chapter 3 Section 3.4 Rational Functions In the last few sections, we have built polynomials based on the positive whole number power functions. In this section we eplore functions based on power functions

### 12. Introduction and Chapter Objectives

Real Analog - Circuits 1 Chapter 1: Steady-State Sinusoidal Power 1. Introduction and Chapter Objectives In this chapter we will address the issue of power transmission via sinusoidal or AC) signals. This

### Graphing Review Part 1: Circles, Ellipses and Lines

Graphing Review Part : Circles, Ellipses and Lines Definition The graph of an equation is the set of ordered pairs, (, y), that satisfy the equation We can represent the graph of a function by sketching

### Section 3.4 Rational Functions

3.4 Rational Functions 93 Section 3.4 Rational Functions In the last few sections, we have built polynomials based on the positive whole number power functions. In this section we eplore functions based

### Transient Response of a Second-Order System

Transient Response of a Second-Order System ECEN 830 Spring 01 1. Introduction In connection with this experiment, you are selecting the gains in your feedback loop to obtain a well-behaved closed-loop

### Frequency Response part 2 (I&N Chap 12)

Frequency Response part 2 (I&N Chap 12) Introduction & TFs Decibel Scale & Bode Plots Resonance Scaling Filter Networks Applications/Design Frequency response; based on slides by J. Yan Slide 3.1 Example

### Capacitor and Inductor

Capacitor and Inductor Copyright (c) 2015 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or

### ECE382/ME482 Spring 2005 Homework 6 Solution April 17, (s/2 + 1) s(2s + 1)[(s/8) 2 + (s/20) + 1]

ECE382/ME482 Spring 25 Homework 6 Solution April 17, 25 1 Solution to HW6 P8.17 We are given a system with open loop transfer function G(s) = 4(s/2 + 1) s(2s + 1)[(s/8) 2 + (s/2) + 1] (1) and unity negative

### Chapter 23: Magnetic Flux and Faraday s Law of Induction

Chapter 3: Magnetic Flux and Faraday s Law of Induction Answers Conceptual Questions 6. Nothing. In this case, the break prevents a current from circulating around the ring. This, in turn, prevents the

### Laboratory III: Operational Amplifiers

Physics 33, Fall 2008 Lab III - Handout Laboratory III: Operational Amplifiers Introduction Operational amplifiers are one of the most useful building blocks of analog electronics. Ideally, an op amp would

### Jim Lambers MAT 460 Fall Semester Lecture 2 Notes

Jim Lambers MAT 460 Fall Semester 2009-10 Lecture 2 Notes These notes correspond to Section 1.1 in the text. Review of Calculus Among the mathematical problems that can be solved using techniques from

### Chapter 10 Introduction to the Derivative

Chapter 0 Introduction to the Derivative The concept of a derivative takes up half the study of Calculus. A derivative, basically, represents rates of change. 0. Limits: Numerical and Graphical Approaches

### The Relation Between the 3-D Bode Diagram and the Root Locus. Insights into the connection between these classical methods. By Panagiotis Tsiotras

F E A T U R E The Relation Between the -D Bode Diagram and the Root Locus Insights into the connection between these classical methods Bode diagrams and root locus plots have been the cornerstone of control

### ECE137B Final Exam. There are 5 problems on this exam and you have 3 hours There are pages 1-19 in the exam: please make sure all are there.

ECE37B Final Exam There are 5 problems on this exam and you have 3 hours There are pages -9 in the exam: please make sure all are there. Do not open this exam until told to do so Show all work: Credit

### I. Impedance of an R-L circuit.

I. Impedance of an R-L circuit. [For inductor in an AC Circuit, see Chapter 31, pg. 1024] Consider the R-L circuit shown in Figure: 1. A current i(t) = I cos(ωt) is driven across the circuit using an AC

### Pre-Calculus Mathematics Limit Process Calculus

NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Mrs. Nguyen s Initial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to find

### Algebra I Calculator Activities

First Nine Weeks SOL Objectives Calculating Measures of Central Tendency SOL A.17 Organize a set of data Calculate the mean, median, mode, and range of a set of data Describe the relationships between

### Designing Information Devices and Systems II Spring 2016 Anant Sahai and Michel Maharbiz Midterm 2

EECS 16B Designing Information Devices and Systems II Spring 2016 Anant Sahai and Michel Maharbiz Midterm 2 Exam location: 145 Dwinelle (SIDs ending in 1 and 5) PRINT your student ID: PRINT AND SIGN your

### AP Physics C: Electricity and Magnetism

2017 AP Physics C: Electricity and Magnetism Scoring Guidelines 2017 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College

Chapter 4 Sinusoidal Steady-State Analysis In this unit, we consider circuits in which the sources are sinusoidal in nature. The review section of this unit covers most of section 9.1 9.9 of the text.

### AP Calculus Worksheet: Chapter 2 Review Part I

AP Calculus Worksheet: Chapter 2 Review Part I 1. Given y = f(x), what is the average rate of change of f on the interval [a, b]? What is the graphical interpretation of your answer? 2. The derivative

### Solving Equations Quick Reference

Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number

### The output voltage is given by,

71 The output voltage is given by, = (3.1) The inductor and capacitor values of the Boost converter are derived by having the same assumption as that of the Buck converter. Now the critical value of the

### CORRESPONDENCE BETWEEN PHASOR TRANSFORMS AND FREQUENCY RESPONSE FUNCTION IN RLC CIRCUITS

CORRESPONDENCE BETWEEN PHASOR TRANSFORMS AND FREQUENCY RESPONSE FUNCTION IN RLC CIRCUITS Hassan Mohamed Abdelalim Abdalla Polytechnic Department of Engineering and Architecture, University of Studies of

### Analogue Filters Design and Simulation by Carsten Kristiansen Napier University. November 2004

Analogue Filters Design and Simulation by Carsten Kristiansen Napier University November 2004 Title page Author: Carsten Kristiansen. Napier No: 04007712. Assignment title: Analogue Filters Design and

### Things to remember: x n a 1. x + a 0. x n + a n-1. P(x) = a n. Therefore, lim g(x) = 1. EXERCISE 3-2

lim f() = lim (0.8-0.08) = 0, " "!10!10 lim f() = lim 0 = 0.!10!10 Therefore, lim f() = 0.!10 lim g() = lim (0.8 - "!10!10 0.042-3) = 1, " lim g() = lim 1 = 1.!10!0 Therefore, lim g() = 1.!10 EXERCISE

### Chapter 3.5: Rational Functions

Chapter.5: Rational Functions A rational number is a ratio of two integers. A rational function is a quotient of two polynomials. All rational numbers are, therefore, rational functions as well. Let s

### Section 6.4 DEs with Discontinuous Forcing Functions

Section 6.4 DEs with Discontinuous Forcing Functions Key terms/ideas: Discontinuous forcing function in nd order linear IVPs Application of Laplace transforms Comparison to viewing the problem s solution

### The Frequency-Response

6 The Frequency-Response Design Method A Perspective on the Frequency-Response Design Method The design of feedback control systems in industry is probably accomplished using frequency-response methods

### ROOT LOCUS. Consider the system. Root locus presents the poles of the closed-loop system when the gain K changes from 0 to. H(s) H ( s) = ( s)

C1 ROOT LOCUS Consider the system R(s) E(s) C(s) + K G(s) - H(s) C(s) R(s) = K G(s) 1 + K G(s) H(s) Root locus presents the poles of the closed-loop system when the gain K changes from 0 to 1+ K G ( s)

### Polynomial Functions. Essential Questions. Module Minute. Key Words. CCGPS Advanced Algebra Polynomial Functions

CCGPS Advanced Algebra Polynomial Functions Polynomial Functions Picture yourself riding the space shuttle to the international space station. You will need to calculate your speed so you can make the

### SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION

2.25 SECTION 2.3: LONG AND SYNTHETIC POLYNOMIAL DIVISION PART A: LONG DIVISION Ancient Example with Integers 2 4 9 8 1 In general: dividend, f divisor, d We can say: 9 4 = 2 + 1 4 By multiplying both sides

### Reteach Multiplying and Dividing Rational Expressions

8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression:

### INTRODUCTION TO DIGITAL CONTROL

ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a linear-time-invariant

### A number that can be written as, where p and q are integers and q Number.

RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.

### 2. Approximate the real zero of f(x) = x3 + x + 1 to the nearest tenth. Answer: Substitute all the values into f(x) and find which is closest to zero

Unit 2 Examples(K) 1. Find all the real zeros of the function. Answer: Simply substitute the values given in all the functions and see which option when substituted, all the values go to zero. That is

### Physics 1308 Exam 2 Summer 2015

Physics 1308 Exam 2 Summer 2015 E2-01 2. The direction of the magnetic field in a certain region of space is determined by firing a test charge into the region with its velocity in various directions in

### y = log b Exponential and Logarithmic Functions LESSON THREE - Logarithmic Functions Lesson Notes Example 1 Graphing Logarithms

y = log b Eponential and Logarithmic Functions LESSON THREE - Logarithmic Functions Eample 1 Logarithmic Functions Graphing Logarithms a) Draw the graph of f() = 2 b) Draw the inverse of f(). c) Show algebraically

### 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

### 12 Chapter Driven RLC Circuits

hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...

### Biquad Filter. by Kenneth A. Kuhn March 8, 2013

by Kenneth A. Kuhn March 8, 201 The biquad filter implements both a numerator and denominator quadratic function in s thus its name. All filter outputs have identical second order denominator in s and

### Massachusetts Tests for Educator Licensure (MTEL )

Massachusetts Tests for Educator Licensure (MTEL ) BOOKLET 2 Mathematics Subtest Copyright 2010 Pearson Education, Inc. or its affiliate(s). All rights reserved. Evaluation Systems, Pearson, P.O. Box 226,

### Feedback design for the Buck Converter

Feedback design for the Buck Converter Portland State University Department of Electrical and Computer Engineering Portland, Oregon, USA December 30, 2009 Abstract In this paper we explore two compensation

### Calculus concepts and applications

Calculus concepts and applications This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

### CIRCUIT ANALYSIS II. (AC Circuits)

Will Moore MT & MT CIRCUIT ANALYSIS II (AC Circuits) Syllabus Complex impedance, power factor, frequency response of AC networks including Bode diagrams, second-order and resonant circuits, damping and

### 7.4 STEP BY STEP PROCEDURE TO DRAW THE ROOT LOCUS DIAGRAM

ROOT LOCUS TECHNIQUE. Values of on the root loci The value of at any point s on the root loci is determined from the following equation G( s) H( s) Product of lengths of vectors from poles of G( s)h( s)

### The First Derivative Test

The First Derivative Test We have already looked at this test in the last section even though we did not put a name to the process we were using. We use a y number line to test the sign of the first derivative

### EECE 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

### Chapter 2 Analysis of Graphs of Functions

Chapter Analysis of Graphs of Functions Chapter Analysis of Graphs of Functions Covered in this Chapter:.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry..

### MAK 391 System Dynamics & Control. Presentation Topic. The Root Locus Method. Student Number: Group: I-B. Name & Surname: Göksel CANSEVEN

MAK 391 System Dynamics & Control Presentation Topic The Root Locus Method Student Number: 9901.06047 Group: I-B Name & Surname: Göksel CANSEVEN Date: December 2001 The Root-Locus Method Göksel CANSEVEN

### I 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

### Electrochemical methods : Fundamentals and Applications

Electrochemical methods : Fundamentals and Applications Lecture Note 7 May 19, 2014 Kwang Kim Yonsei University kbkim@yonsei.ac.kr 39 8 7 34 53 Y O N Se I 88.91 16.00 14.01 78.96 126.9 Electrochemical

### LCR Series Circuits. AC Theory. Introduction to LCR Series Circuits. Module. What you'll learn in Module 9. Module 9 Introduction

Module 9 AC Theory LCR Series Circuits Introduction to LCR Series Circuits What you'll learn in Module 9. Module 9 Introduction Introduction to LCR Series Circuits. Section 9.1 LCR Series Circuits. Amazing

### Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits. Nov. 7 & 9, 2011

Lectures 16 & 17 Sinusoidal Signals, Complex Numbers, Phasors, Impedance & AC Circuits Nov. 7 & 9, 2011 Material from Textbook by Alexander & Sadiku and Electrical Engineering: Principles & Applications,

### CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic

CHAPTER EIGHT: SOLVING QUADRATIC EQUATIONS Review April 9 Test April 17 The most important equations at this level of mathematics are quadratic equations. They can be solved using a graph, a perfect square,

### (b) A unity feedback system is characterized by the transfer function. Design a suitable compensator to meet the following specifications:

1. (a) The open loop transfer function of a unity feedback control system is given by G(S) = K/S(1+0.1S)(1+S) (i) Determine the value of K so that the resonance peak M r of the system is equal to 1.4.

### INTRODUCTION ELECTROSTATIC POTENTIAL ENERGY. Introduction. Electrostatic potential energy. Electric potential. for a system of point charges

Chapter 4 ELECTRIC POTENTIAL Introduction Electrostatic potential energy Electric potential for a system of point charges for a continuous charge distribution Why determine electic potential? Determination

### p324 Section 5.2: The Natural Logarithmic Function: Integration

p324 Section 5.2: The Natural Logarithmic Function: Integration Theorem 5.5: Log Rule for Integration Let u be a differentiable function of x 1. 2. Example 1: Using the Log Rule for Integration ** Note:

### Math 1320, Section 10 Quiz IV Solutions 20 Points

Math 1320, Section 10 Quiz IV Solutions 20 Points Please answer each question. To receive full credit you must show all work and give answers in simplest form. Cell phones and graphing calculators are

### Basic Equation Solving Strategies

Basic Equation Solving Strategies Case 1: The variable appears only once in the equation. (Use work backwards method.) 1 1. Simplify both sides of the equation if possible.. Apply the order of operations

### Representation of standard models: Transfer functions are often used to represent standard models of controllers and signal filters.

Chapter 5 Transfer functions 5.1 Introduction Transfer functions is a model form based on the Laplace transform, cf. Chapter 4. Transfer functions are very useful in analysis and design of linear dynamic

### Exact 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

### Fourier series. XE31EO2 - Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2 - Pavel Máša - Fourier Series

Fourier series Electrical Circuits Lecture - Fourier Series Filtr RLC defibrillator MOTIVATION WHAT WE CAN'T EXPLAIN YET Source voltage rectangular waveform Resistor voltage sinusoidal waveform - Fourier

### Polynomial Functions of Higher Degree

SAMPLE CHAPTER. NOT FOR DISTRIBUTION. 4 Polynomial Functions of Higher Degree Polynomial functions of degree greater than 2 can be used to model data such as the annual temperature fluctuations in Daytona

### Course. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.

Maclaurin Series Learning Outcomes After reading this theory sheet, you should recognise the difference between a function and its polynomial epansion (if it eists!) understand what is meant by a series

### Tangent Lines and Derivatives

The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the

### Contents Partial Fraction Theory Real Quadratic Partial Fractions Simple Roots Multiple Roots The Sampling Method The Method of Atoms Heaviside s

Contents Partial Fraction Theory Real Quadratic Partial Fractions Simple Roots Multiple Roots The Sampling Method The Method of Atoms Heaviside s Coverup Method Extension to Multiple Roots Special Methods

### Advanced Hydraulics Dr. Suresh A. Kartha Department of Civil Engineering Indian Institute of Technology, Guwahati

Advanced Hydraulics Dr. Suresh A. Kartha Department of Civil Engineering Indian Institute of Technology, Guwahati Module - 3 Varied Flows Lecture - 7 Gradually Varied Flow Computations Part 1 (Refer Slide

### How do physicists study problems?

What is Physics? The branch of science that studies the physical world (from atoms to the universe); The study of the nature of matter and energy and how they are related; The ability to understand or

### To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function

### Chapter 3. Loop and Cut-set Analysis

Chapter 3. Loop and Cut-set Analysis By: FARHAD FARADJI, Ph.D. Assistant Professor, Electrical Engineering, K.N. Toosi University of Technology http://wp.kntu.ac.ir/faradji/electriccircuits2.htm References:

### EE 435. Lecture 2: Basic Op Amp Design. - Single Stage Low Gain Op Amps

EE 435 ecture 2: Basic Op Amp Design - Single Stage ow Gain Op Amps 1 Review from last lecture: How does an amplifier differ from an operational amplifier?? Op Amp Amplifier Amplifier used in open-loop

### Part D: Kinematic Graphing - ANSWERS

Part D: Kinematic Graphing - ANSWERS 31. On the position-time graph below, sketch a plot representing the motion of an object which is.... Label each line with the corresponding letter (e.g., "a", "b",

### AC Circuit Analysis and Measurement Lab Assignment 8

Electric Circuit Lab Assignments elcirc_lab87.fm - 1 AC Circuit Analysis and Measurement Lab Assignment 8 Introduction When analyzing an electric circuit that contains reactive components, inductors and

### Lab 5 CAPACITORS & RC CIRCUITS

L051 Name Date Partners Lab 5 CAPACITORS & RC CIRCUITS OBJECTIVES OVERVIEW To define capacitance and to learn to measure it with a digital multimeter. To explore how the capacitance of conducting parallel

### Physics 2112 Unit 19

Physics 11 Unit 19 Today s oncepts: A) L circuits and Oscillation Frequency B) Energy ) RL circuits and Damping Electricity & Magnetism Lecture 19, Slide 1 Your omments differential equations killing me.