# Analysis and Design of Analog Integrated Circuits Lecture 12. Feedback

Size: px
Start display at page:

Transcription

1 Analysis and Design of Analog Integrated Circuits Lecture 12 Feedback Michael H. Perrott March 11, 2012 Copyright 2012 by Michael H. Perrott All rights reserved.

2 Open Loop Versus Closed Loop Amplifier Topologies Open Loop Closed Loop Z f Source Source V src Z src Amp V src Z src Amp / / (db) (db) w bw w 1 w 2 w bw w 1 w 2 Open loop want all bandwidth limiting poles to be as high in frequency as possible Closed loop want one pole to be dominant and all other parasitic poles to be as high in frequency as possible 2

3 Consider an Open Loop Integrator w unity s / (db) w unity Parameterize integrator in terms of its unity gain frequency, w unity rad/s - Define = w unity /s - Note that H(w unity ) = 1 3

4 Now Surround the Integrator with a Feedback Path Z 2 20log /( -V x ) (db) Z 1 V x w unity The feedforward path is = w unity /s The feedback path is formed by Z 1 and Z 2 Derivation of closed loop transfer function: V x Z 2 = V x 0 Z 1 and = ( V x ) = 1+Z 1 /(Z 1 + Z 2 ) 4

5 Observations of Impact of Feedback 20log / Z 2 Z 1 V x 20log(1/β) w bw w unity Define = Z 1 /(Z 1 +Z 2 ) and rewrite transfer function as: = 1+Z 1 /(Z 1 + Z 2 ) = 1+β At low frequencies: At high frequencies: = s 0 1+β = 1 β = s (since β 1) 5

6 General View of Feedback 20log / Z 2 Z 1 V x 20log(1/β) β w bw w unity Closed loop transfer function: = V - in 1+β This is called Black s formula At low frequencies: At high frequencies: = 1 = s 0 β s 6

7 General Observations of Feedback 20log / β 20log(1/β) = 1+β w bw w unity The feedback path sets the closed loop gain at low frequencies - Assumes the open loop gain is large at low frequencies - Implies that accurate closed loop gain can be achieved at low frequencies despite variations in open loop gain The feedback path also influences the closed loop bandwidth 7

8 Gain Bandwidth Product for Closed Loop Systems 20log / = β 1+β The low frequency gain is: 20log(1/β) w bw s 0 = 1 β w unity The bandwidth roughly corresponds to: H(w bw ) 1 β w unity For = w unity w bw 1 β w unity = 1 β w bw s Closed loop systems exhibit constant gain-bandwidth product set by the unity gain frequency of the open loop amplifier 8

9 Example: Unity Gain Amplifier 20log / β = 1 = 1+ The low frequency gain is: s 0 =1 The bandwidth roughly corresponds to: w unity H(w bw ) w unity Gain bandwidth product is w unity : 1 w unity = w unity Unity gain closed loop amplifiers maximize the closed loop bandwidth assuming closed loop gain 1 9

10 Issue: Open Loop Amplifiers have Finite DC Gain 20log / 20log(K) β = 1 = 1+ Let us now model as: = w dominant w unity To first order, the closed loop bandwidth and gain are relatively unchanged K 1+s/w dominant What is the impact of having finite, open loop, DC gain? 10

11 Further Examination of Finite, Open Loop, DC Gain 20log / 20log(K) = K 1+s/w dominant β = 1 = 1+ w dominant w unity For unity gain configuration of closed loop amplifier: = = K s 0 1+ s 0 1+K = 1 1+1/K We see that finite open loop DC gain leads to a slight reduction of the closed loop DC gain - We want K >> 1 for the unity gain closed loop amplifier 11

12 More General View of Finite, Open Loop, DC Gain 20log / 20log(K) = K 1+s/w dominant β 20log(1/β) = 1+β w bw w unity For general configuration of closed loop amplifier: µ K 1 = s 0 1+β K = 1 β 1+(1/β)/K Finite open loop DC gain still leads to reduction of closed loop DC gain - We want K >> 1/ in this case - We will see implications of this issue later in the class 12

13 The Issue of Parasitic Open Loop Poles 20log / 20log(K) β 20log(1/β) = 1+β w bw w p Practical amplifiers have non-dominant poles, too: µ µ K 1 = 1+s/w dominant 1+s/w p - Of course, there can be multiple parasitic poles and also zeros A key issue of such parasitic poles is their influence on the stability of the closed loop amplifier 13

14 Key Tool for Assessing Stability: Open Loop Response 20log / 20log(K) We define the open loop response, A(s), as: A(s) =β Note that the unity gain frequency, w 0, of A(w) is approximately the same as the closed loop bandwidth, w bw = β 1+β 20log(1/β) A(w - 0 ) =1 β H(w 0 ) =1 H(w 0 ) =1/β Looking at the plot above, we can see that the intersection of and 1/ corresponds to the closed loop bandwidth, w bw w bw w p 14

15 Stability Analysis Based on Phase Margin of A(w) Phase margin is a key metric when examining the stability of a system - Phase margin is defined as phase{a(w 0 )} w 0 corresponds to the unity gain frequency of the open loop response (i.e., A(w 0 ) = 1) w 0 is approximately the same as the closed loop bandwidth, w - bw Phase margin must be greater than 0 degrees for the closed loop system to be stable Typically want phase margin to be greater than 45 Key skill: you must be able to plot Bode plots in both magnitude and phase! 15

16 Review of Bode Plot Basics Example: A(w) = - Log of magnitude (db): 1+jw/w z (1 + jw/w p1 )(1 + jw/w p2 ) 20 log A(w) =20log 1+jw/w z 20 log 1+jw/w p1 20 log 1+jw/w p2 Taking the log allows the poles and zeros to be plotted separately and then added together - Phase: 6 A(w) = 6 (1 + jw/w z ) 6 (1 + jw/w p1 ) 6 (1 + jw/w p2 ) Phase of poles and zeros can also be plotted separately and then added together 16

17 Review: Plotting the Magnitude of Poles Plot the magnitude response of pole w p1 20 log A p1 (w) =20log 1 1+jw/w p1 = 20 log 1+jw/w p1 - For w << w p1 : - For w >> w p1 : 20 log A p1 (w) 20 log 1 =0 20 log A p1 (w) 20 log w/w p1 20log A p1 (ω) ω ω p1-2/decade 17

18 Plotting the Phase of Poles Plot the phase response of pole w p1 6 A p1 (w) = 6 (1 + jw/w p1 )= arctan (w/w p1 ) - For w << w p1 : - For w = w p1 : - For w >> w p1 : 6 A p1 (w) arctan (0) = 0 6 A p1 (w) arctan (1) = 45 6 A p1 (w) arctan ( ) = 90 A p1 (ω) ω ω 0 o p1 /10 p1-45 o ω p1 10 ω -90 o 18

19 Review: Plotting the Magnitude of Zeros Plot the magnitude response of zero w z 20 log A z (w) =20log 1+jw/w z - For w << w z : - For w >> w z : 20 log A z (w) 20 log 1 =0 20 log A z (w) 20 log w/w z 20log A z (ω) 2/decade ω ω z 19

20 Plotting the Phase of Zeros Plot the phase response of zero w z 6 A z (w) =6 (1 + jw/w z )=arctan(w/w z ) - For w << w z : - For w = w z : - For w >> w z : 6 A z (w) arctan (0) = 0 6 A z (w) arctan (1) = 45 6 A z (w) arctan ( ) =90 90 o A z (ω) 45 o 0 o ω z /10 ω z ω z 10 ω 20

21 Example of Closed Loop Stability Evaluation 20log / β 20log(1/β) = 1+β Consider the case where: µ µ K 1 = s 1+s/w p w bw w p - This implies that: A(s) =β µ K s µ 1 1+s/w p 21

22 Phase Margin Versus Open Loop Gain Open loop gain increased 20log A(f) Evaluation of Phase Margin Dominant pole pair Closed Loop Pole Locations C B Im{s} -90 o angle(a(f)) f p C B A f A 0 Re{s} A -120 o PM = 59 o for A -150 o PM = 45 o for B PM = 33 o for C B -180 o C Note the closed loop pole locations versus open loop gain - Is the closed loop system unstable for any case above? 22

23 Corresponding Closed Loop Behavior Closed Loop Frequency Response Closed Loop Step Response 5 db -5 db A B C C B A 0.6 Frequency response sees more peaking with higher open loop gain - How does this relate to the movement of the closed loop pole locations? Step response see more ringing with higher open loop gain f p f - How does this relate to the closed loop frequency response? 0 t 23

24 Some Key Observations 20log / β 20log(1/β) = 1+β A(s) =β w bw w p We have seen that increasing the open loop gain of A(w) leads to higher closed loop bandwidth - How is this consistent with the statement that increasing closed loop gain leads to lower closed loop bandwidth? As an exercise, consider the impact of the following: - Keep unchanged and increase the open loop gain of - Keep unchanged and increase 24

25 Example 2 of Closed Loop Stability Evaluation 20log / = β 1+β 20log(1/β) Consider the case where: µ µ K = s - This implies that: µ µ K A(s) =β s w bw 1 1+s/w p1 1 1+s/w p2 1 1+s/w p3 w p1, w p2, w p3 1 1+s/w p1 1 1+s/w p2 1 1+s/w p3 25

26 Phase Margin Versus Open Loop Gain Open loop gain increased -90 o -165 o -180 o 20log A(f) angle(a(f)) Evaluation of Phase Margin f p1 f p2 f p3 C B A f PM = 72 o for A PM = 51 o for B PM = -12 o for C Closed Loop Pole Locations Non-dominant poles Dominant pole pair A A B B Im{s} C Re{s} o -315 o C Note the closed loop pole locations versus open loop gain - Is the closed loop system unstable for any case above? 26

27 Corresponding Closed Loop Behavior Closed Loop Frequency Response Closed Loop Step Response C A B C 1 B A Frequency Time Frequency response again sees more peaking with higher open loop gain - How does this relate to the movement of the closed loop pole locations? Step response ringing grows for high open loop gain - How does this relate to the closed loop pole locations? 27

28 Open Loop Versus Closed Loop Amplifier Topologies Open Loop Closed Loop Z f Source Source V src Z src Amp V src Z src Amp / / (db) (db) w bw w 1 w 2 w bw w 1 w 2 Now that we understand the phase margin criterion, can you explain why amplifiers designed to be within a closed loop system should have one dominant pole that is much lower in frequency than the parasitic poles? 28

29 Summary Feedback systems offer the benefit of accurate gain at low frequencies - Assumes accurate feedback and high open loop DC gain - Gain-bandwidth product of the closed loop system equals w unity of the open loop amplifier Accuracy of the closed loop DC gain is reduced with lower open loop DC gain - Want the open loop DC gain to be much higher than the desired closed loop DC gain for reasonable accuracy Stability of the closed loop system is often evaluated using the phase margin criterion - Examines the phase at unity gain frequency of the open loop response, A(w 0 ) = H(w 0 ), where A(w 0 ) = 1 w 0 is approximately the same as the closed loop bandwidth, w bw 29

### 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

### 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

### Electronics II. Final Examination

f3fs_elct7.fm - The University of Toledo EECS:3400 Electronics I Section Student Name Electronics II Final Examination Problems Points.. 3 3. 5 Total 40 Was the exam fair? yes no Analog Electronics f3fs_elct7.fm

### DESIGN MICROELECTRONICS ELCT 703 (W17) LECTURE 3: OP-AMP CMOS CIRCUIT. Dr. Eman Azab Assistant Professor Office: C

MICROELECTRONICS ELCT 703 (W17) LECTURE 3: OP-AMP CMOS CIRCUIT DESIGN Dr. Eman Azab Assistant Professor Office: C3.315 E-mail: eman.azab@guc.edu.eg 1 TWO STAGE CMOS OP-AMP It consists of two stages: First

### Exercises for lectures 13 Design using frequency methods

Exercises for lectures 13 Design using frequency methods Michael Šebek Automatic control 2016 31-3-17 Setting of the closed loop bandwidth At the transition frequency in the open loop is (from definition)

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect

### Chapter 10 Feedback. PART C: Stability and Compensation

1 Chapter 10 Feedback PART C: Stability and Compensation Example: Non-inverting Amplifier We are analyzing the two circuits (nmos diff pair or pmos diff pair) to realize this symbol: either of the circuits

### Lecture 17 Date:

Lecture 17 Date: 27.10.2016 Feedback and Properties, Types of Feedback Amplifier Stability Gain and Phase Margin Modification Elements of Feedback System: (a) The feed forward amplifier [H(s)] ; (b) A

### 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

### Advanced Analog Integrated Circuits. Operational Transconductance Amplifier II Multi-Stage Designs

Advanced Analog Integrated Circuits Operational Transconductance Amplifier II Multi-Stage Designs Bernhard E. Boser University of California, Berkeley boser@eecs.berkeley.edu Copyright 2016 by Bernhard

### 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

### ECEN 326 Electronic Circuits

ECEN 326 Electronic Circuits Stability Dr. Aydın İlker Karşılayan Texas A&M University Department of Electrical and Computer Engineering Ideal Configuration V i Σ V ε a(s) V o V fb f a(s) = V o V ε (s)

### EE 508 Lecture 4. Filter Concepts/Terminology Basic Properties of Electrical Circuits

EE 58 Lecture 4 Filter Concepts/Terminology Basic Properties of Electrical Circuits Review from Last Time Filter Design Process Establish Specifications - possibly T D (s) or H D (z) - magnitude and phase

### Boise State University Department of Electrical Engineering ECE461 Control Systems. Control System Design in the Frequency Domain

Boise State University Department of Electrical Engineering ECE6 Control Systems Control System Design in the Frequency Domain Situation: Consider the following block diagram of a type- servomechanism:

### EE C128 / ME C134 Fall 2014 HW 8 - Solutions. HW 8 - Solutions

EE C28 / ME C34 Fall 24 HW 8 - Solutions HW 8 - Solutions. Transient Response Design via Gain Adjustment For a transfer function G(s) = in negative feedback, find the gain to yield a 5% s(s+2)(s+85) overshoot

### 1 (20 pts) Nyquist Exercise

EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. For each G(s), hand sketch the Nyquist diagram, determine Z = P N, algebraically

### Frequency methods for the analysis of feedback systems. Lecture 6. Loop analysis of feedback systems. Nyquist approach to study stability

Lecture 6. Loop analysis of feedback systems 1. Motivation 2. Graphical representation of frequency response: Bode and Nyquist curves 3. Nyquist stability theorem 4. Stability margins Frequency methods

### Lecture 50 Changing Closed Loop Dynamic Response with Feedback and Compensation

Lecture 50 Changing Closed Loop Dynamic Response with Feedback and Compensation 1 A. Closed Loop Transient Response Waveforms 1. Standard Quadratic T(s) Step Response a. Q > 1/2 Oscillatory decay to a

### ECEN 607 (ESS) Op-Amps Stability and Frequency Compensation Techniques. Analog & Mixed-Signal Center Texas A&M University

ECEN 67 (ESS) Op-Amps Stability and Frequency Compensation Techniques Analog & Mixed-Signal Center Texas A&M University Stability of Linear Systems Harold S. Black, 97 Negative feedback concept Negative

### FEEDBACK AND STABILITY

FEEDBCK ND STBILITY THE NEGTIVE-FEEDBCK LOOP x IN X OUT x S + x IN x OUT Σ Signal source _ β Open loop Closed loop x F Feedback network Output x S input signal x OUT x IN x F feedback signal x IN x S x

### Homework Assignment 11

Homework Assignment Question State and then explain in 2 3 sentences, the advantage of switched capacitor filters compared to continuous-time active filters. (3 points) Continuous time filters use resistors

### ECE 486 Control Systems

ECE 486 Control Systems Spring 208 Midterm #2 Information Issued: April 5, 208 Updated: April 8, 208 ˆ This document is an info sheet about the second exam of ECE 486, Spring 208. ˆ Please read the following

### 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

### 6.302 Feedback Systems

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.302 Feedback Systems Fall Term 2005 Issued : November 18, 2005 Lab 2 Series Compensation in Practice Due

### Today (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10

Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:

### Homework 6 Solutions and Rubric

Homework 6 Solutions and Rubric EE 140/40A 1. K-W Tube Amplifier b) Load Resistor e) Common-cathode a) Input Diff Pair f) Cathode-Follower h) Positive Feedback c) Tail Resistor g) Cc d) Av,cm = 1/ Figure

### Frequency Response Techniques

4th Edition T E N Frequency Response Techniques SOLUTION TO CASE STUDY CHALLENGE Antenna Control: Stability Design and Transient Performance First find the forward transfer function, G(s). Pot: K 1 = 10

### 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.

### Frequency response. Pavel Máša - XE31EO2. XE31EO2 Lecture11. Pavel Máša - XE31EO2 - Frequency response

Frequency response XE3EO2 Lecture Pavel Máša - Frequency response INTRODUCTION Frequency response describe frequency dependence of output to input voltage magnitude ratio and its phase shift as a function

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 24: Compensation in the Frequency Domain Overview In this Lecture, you will learn: Lead Compensators Performance Specs Altering

### CDS 101/110a: Lecture 8-1 Frequency Domain Design

CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve

### r + - FINAL June 12, 2012 MAE 143B Linear Control Prof. M. Krstic

MAE 43B Linear Control Prof. M. Krstic FINAL June, One sheet of hand-written notes (two pages). Present your reasoning and calculations clearly. Inconsistent etchings will not be graded. Write answers

### Exercise 1 (A Non-minimum Phase System)

Prof. Dr. E. Frazzoli 5-59- Control Systems I (Autumn 27) Solution Exercise Set 2 Loop Shaping clruch@ethz.ch, 8th December 27 Exercise (A Non-minimum Phase System) To decrease the rise time of the system,

### I. Frequency Response of Voltage Amplifiers

I. Frequency Response of Voltage Amplifiers A. Common-Emitter Amplifier: V i SUP i OUT R S V BIAS R L v OUT V Operating Point analysis: 0, R s 0, r o --->, r oc --->, R L ---> Find V BIAS such that I C

### OPERATIONAL AMPLIFIER APPLICATIONS

OPERATIONAL AMPLIFIER APPLICATIONS 2.1 The Ideal Op Amp (Chapter 2.1) Amplifier Applications 2.2 The Inverting Configuration (Chapter 2.2) 2.3 The Non-inverting Configuration (Chapter 2.3) 2.4 Difference

### Advanced Analog Integrated Circuits. Operational Transconductance Amplifier I & Step Response

Advanced Analog Integrated Circuits Operational Transconductance Amplifier I & Step Response Bernhard E. Boser University of California, Berkeley boser@eecs.berkeley.edu Copyright 2016 by Bernhard Boser

### Stability & Compensation

Advanced Analog Building Blocks Stability & Compensation Wei SHEN (KIP) 1 Bode Plot real zeros zeros with complex conjugates real poles poles with complex conjugates http://lpsa.swarthmore.edu/bode/bode.html

### B. T(s) Modification Design Example 1. DC Conditions 2. Open Loop AC Conditions 3. Closed Loop Conditions

Lecture 51 Tailoring Dynamic Response with Compensation A. Compensation Networks 1. Overview of G c (Alterations and Tailoring) a. Crude Single Pole G C Tailoring b. Elegant Double Pole/Double Zero G C

### ECEN 325 Electronics

ECEN 325 Electronics Operational Amplifiers Dr. Aydın İlker Karşılayan Texas A&M University Department of Electrical and Computer Engineering Opamp Terminals positive supply inverting input terminal non

### Stability of CL System

Stability of CL System Consider an open loop stable system that becomes unstable with large gain: At the point of instability, K( j) G( j) = 1 0dB K( j) G( j) K( j) G( j) K( j) G( j) =± 180 o 180 o Closed

### ESE319 Introduction to Microelectronics. Feedback Basics

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

### 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

### Digital Control Systems

Digital Control Systems Lecture Summary #4 This summary discussed some graphical methods their use to determine the stability the stability margins of closed loop systems. A. Nyquist criterion Nyquist

### Compensation 8. f4 that separate these regions of stability and instability. The characteristic S 0 L U T I 0 N S

S 0 L U T I 0 N S Compensation 8 Note: All references to Figures and Equations whose numbers are not preceded by an "S"refer to the textbook. As suggested in Lecture 8, to perform a Nyquist analysis, we

### Robust and Optimal Control, Spring A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization

Robust and Optimal Control, Spring 2015 Instructor: Prof. Masayuki Fujita (S5-303B) A: SISO Feedback Control A.1 Internal Stability and Youla Parameterization A.2 Sensitivity and Feedback Performance A.3

### Exercise 1 (A Non-minimum Phase System)

Prof. Dr. E. Frazzoli 5-59- Control Systems I (HS 25) Solution Exercise Set Loop Shaping Noele Norris, 9th December 26 Exercise (A Non-minimum Phase System) To increase the rise time of the system, we

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using

### Bandwidth of op amps. R 1 R 2 1 k! 250 k!

Bandwidth of op amps An experiment - connect a simple non-inverting op amp and measure the frequency response. From the ideal op amp model, we expect the amp to work at any frequency. Is that what happens?

### 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

### Lectures on STABILITY

University of California Berkeley College of Engineering Department of Electrical Engineering and Computer Science νin ( ) Effect of Feedback on Frequency Response a SB Robert W. Brodersen EECS40 Analog

### EE3CL4: Introduction to Linear Control Systems

1 / 30 EE3CL4: Introduction to Linear Control Systems Section 9: of and using Techniques McMaster University Winter 2017 2 / 30 Outline 1 2 3 4 / 30 domain analysis Analyze closed loop using open loop

### Theory of Machines and Automatic Control Winter 2018/2019

Theory of Machines and Automatic Control Winter 2018/2019 Lecturer: Sebastian Korczak, PhD, Eng. Institute of Machine Design Fundamentals - Department of Mechanics http://www.ipbm.simr.pw.edu.pl/ Lecture

### R10 JNTUWORLD B 1 M 1 K 2 M 2. f(t) Figure 1

Code No: R06 R0 SET - II B. Tech II Semester Regular Examinations April/May 03 CONTROL SYSTEMS (Com. to EEE, ECE, EIE, ECC, AE) Time: 3 hours Max. Marks: 75 Answer any FIVE Questions All Questions carry

### Electronics II. Final Examination

The University of Toledo f6fs_elct7.fm - Electronics II Final Examination Problems Points. 5. 0 3. 5 Total 40 Was the exam fair? yes no The University of Toledo f6fs_elct7.fm - Problem 5 points Given is

### R a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies.

SET - 1 II B. Tech II Semester Supplementary Examinations Dec 01 1. a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies..

### CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version

CONTROL SYSTEMS ENGINEERING Sixth Edition International Student Version Norman S. Nise California State Polytechnic University, Pomona John Wiley fir Sons, Inc. Contents PREFACE, vii 1. INTRODUCTION, 1

### Active Control? Contact : Website : Teaching

Active Control? Contact : bmokrani@ulb.ac.be Website : http://scmero.ulb.ac.be Teaching Active Control? Disturbances System Measurement Control Controler. Regulator.,,, Aims of an Active Control Disturbances

### 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

### Automatic Control (TSRT15): Lecture 7

Automatic Control (TSRT15): Lecture 7 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13-282226 Office: B-house extrance 25-27 Outline 2 Feedforward

### Chapter 6 Frequency response of circuits. Stability

Chapter 6 Frequency response of circuits. Stability 6.. The frequency response of elementary functions 6... The frequency bandwidth 6... The frequency bandwidth /A/(dB ) A 0 3dB min max 6... The frequency

### (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.

### EE 4343/ Control System Design Project LECTURE 10

Copyright S. Ikenaga 998 All rights reserved EE 4343/5329 - Control System Design Project LECTURE EE 4343/5329 Homepage EE 4343/5329 Course Outline Design of Phase-lead and Phase-lag compensators using

### Electronics II. Midterm II

The University of Toledo su7ms_elct7.fm - Electronics II Midterm II Problems Points. 7. 7 3. 6 Total 0 Was the exam fair? yes no The University of Toledo su7ms_elct7.fm - Problem 7 points Equation (-)

### 1 (s + 3)(s + 2)(s + a) G(s) = C(s) = K P + K I

MAE 43B Linear Control Prof. M. Krstic FINAL June 9, Problem. ( points) Consider a plant in feedback with the PI controller G(s) = (s + 3)(s + )(s + a) C(s) = K P + K I s. (a) (4 points) For a given constant

### Lecture 120 Compensation of Op Amps-I (1/30/02) Page ECE Analog Integrated Circuit Design - II P.E. Allen

Lecture 20 Compensation of Op AmpsI (/30/02) Page 20 LECTURE 20 COMPENSATION OF OP AMPS I (READING: GHLM 425434 and 624638, AH 249260) INTRODUCTION The objective of this presentation is to present the

### ECE382/ME482 Spring 2005 Homework 8 Solution December 11,

ECE382/ME482 Spring 25 Homework 8 Solution December 11, 27 1 Solution to HW8 P1.21 We are given a system with open loop transfer function G(s) = K s(s/2 + 1)(s/6 + 1) and unity negative feedback. We are

### Systems Analysis and Control

Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using the

### Final Exam. 55:041 Electronic Circuits. The University of Iowa. Fall 2013.

Final Exam Name: Max: 130 Points Question 1 In the circuit shown, the op-amp is ideal, except for an input bias current I b = 1 na. Further, R F = 10K, R 1 = 100 Ω and C = 1 μf. The switch is opened at

### Stability of Operational amplifiers

Stability o Operational ampliiers Willy Sansen KULeuven, ESAT-MICAS Leuven, Belgium willy.sansen@esat.kuleuven.be Willy Sansen 0-05 05 Table o contents Use o operational ampliiers Stability o 2-stage opamp

### Chapter 9: Controller design

Chapter 9. Controller Design 9.1. Introduction 9.2. Effect of negative feedback on the network transfer functions 9.2.1. Feedback reduces the transfer function from disturbances to the output 9.2.2. Feedback

### Module 5: Design of Sampled Data Control Systems Lecture Note 8

Module 5: Design of Sampled Data Control Systems Lecture Note 8 Lag-lead Compensator When a single lead or lag compensator cannot guarantee the specified design criteria, a laglead compensator is used.

### CONTROL * ~ SYSTEMS ENGINEERING

CONTROL * ~ SYSTEMS ENGINEERING H Fourth Edition NormanS. Nise California State Polytechnic University, Pomona JOHN WILEY& SONS, INC. Contents 1. Introduction 1 1.1 Introduction, 2 1.2 A History of Control

### MAS107 Control Theory Exam Solutions 2008

MAS07 CONTROL THEORY. HOVLAND: EXAM SOLUTION 2008 MAS07 Control Theory Exam Solutions 2008 Geir Hovland, Mechatronics Group, Grimstad, Norway June 30, 2008 C. Repeat question B, but plot the phase curve

### 6.003: Signals and Systems

6.003: Signals and Systems CT Feedback and Control October 20, 2011 1 Mid-term Examination #2 Wednesday, October 26, 7:30-9:30pm, No recitations on the day of the exam. Coverage: Lectures 1 12 Recitations

### STABILITY OF CLOSED-LOOP CONTOL SYSTEMS

CHBE320 LECTURE X STABILITY OF CLOSED-LOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 10-1 Road Map of the Lecture X Stability of closed-loop control

### MAE 143B - Homework 9

MAE 143B - Homework 9 7.1 a) We have stable first-order poles at p 1 = 1 and p 2 = 1. For small values of ω, we recover the DC gain K = lim ω G(jω) = 1 1 = 2dB. Having this finite limit, our straight-line

### ELECTRONIC SYSTEMS. Basic operational amplifier circuits. Electronic Systems - C3 13/05/ DDC Storey 1

Electronic Systems C3 3/05/2009 Politecnico di Torino ICT school Lesson C3 ELECTONIC SYSTEMS C OPEATIONAL AMPLIFIES C.3 Op Amp circuits» Application examples» Analysis of amplifier circuits» Single and

### Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: Student ID number... Signature...

Automatic Control (MSc in Mechanical Engineering) Lecturer: Andrea Zanchettin Date: 29..23 Given and family names......................solutions...................... Student ID number..........................

### Control Systems I. Lecture 9: The Nyquist condition

Control Systems I Lecture 9: The Nyquist condition adings: Guzzella, Chapter 9.4 6 Åstrom and Murray, Chapter 9.1 4 www.cds.caltech.edu/~murray/amwiki/index.php/first_edition Emilio Frazzoli Institute

### Lecture 15 Nyquist Criterion and Diagram

Lecture Notes of Control Systems I - ME 41/Analysis and Synthesis of Linear Control System - ME86 Lecture 15 Nyquist Criterion and Diagram Department of Mechanical Engineering, University Of Saskatchewan,

### Advanced Current Mirrors and Opamps

Advanced Current Mirrors and Opamps David Johns and Ken Martin (johns@eecg.toronto.edu) (martin@eecg.toronto.edu) slide 1 of 26 Wide-Swing Current Mirrors I bias I V I in out out = I in V W L bias ------------

### Pipelined multi step A/D converters

Department of Electrical Engineering Indian Institute of Technology, Madras Chennai, 600036, India 04 Nov 2006 Motivation for multi step A/D conversion Flash converters: Area and power consumption increase

### Application Report. Mixed Signal Products SLOA021

Application Report May 1999 Mixed Signal Products SLOA021 IMPORTANT NOTICE Texas Instruments and its subsidiaries (TI) reserve the right to make changes to their products or to discontinue any product

### LINEAR CONTROL SYSTEMS. Ali Karimpour Associate Professor Ferdowsi University of Mashhad

LINEAR CONTROL SYSTEMS Ali Karimpour Associate Professor Ferdowsi University of Mashhad Controller design in the frequency domain Topics to be covered include: Lag controller design 2 Dr. Ali Karimpour

### Lecture 7: Two-Ports (2)

Lecture 7: Two-Ports (2) Amin Arbabian Jan M. Rabaey EE142 Fall 2010 Sept. 16 th, 2010 Announcements HW3 Posted, due Thursday September 23 rd, 330pm EE142 drop box. Office hours Two-Port notes by Professor

### Modeling and System Identification for a DC Servo

Modeling and System Identification for a DC Servo Kevin M. Passino and Nicanor Quijano Dept. Electrical Engineering, The Ohio State University 5 Neil Avenue, Columbus, OH 3-7 March 7, Abstract First, you

### 100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) =

1 AME 3315; Spring 215; Midterm 2 Review (not graded) Problems: 9.3 9.8 9.9 9.12 except parts 5 and 6. 9.13 except parts 4 and 5 9.28 9.34 You are given the transfer function: G(s) = 1) Plot the bode plot

### EE 435. Lecture 16. Compensation Systematic Two-Stage Op Amp Design

EE 435 Lecture 6 Compensation Systematic Two-Stage Op Amp Design Review from last lecture Review of Basic Concepts Pole Locations and Stability Theorem: A system is stable iff all closed-loop poles lie

### (a) Find the transfer function of the amplifier. Ans.: G(s) =

126 INTRDUCTIN T CNTR ENGINEERING 10( s 1) (a) Find the transfer function of the amplifier. Ans.: (. 02s 1)(. 001s 1) (b) Find the expected percent overshoot for a step input for the closed-loop system

### 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

### Radar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.

Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Department, Middle East Technical University Armature controlled dc motor Outside θ D output Inside θ r input r θ m Gearbox Control Transmitter

### CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION

CHAPTER 7 : BODE PLOTS AND GAIN ADJUSTMENTS COMPENSATION Objectives Students should be able to: Draw the bode plots for first order and second order system. Determine the stability through the bode plots.

### EE 508 Lecture 24. Sensitivity Functions - Predistortion and Calibration

EE 508 Lecture 24 Sensitivity Functions - Predistortion and Calibration Review from last time Sensitivity Comparisons Consider 5 second-order lowpass filters (all can realize same T(s) within a gain factor)

### PM diagram of the Transfer Function and its use in the Design of Controllers

PM diagram of the Transfer Function and its use in the Design of Controllers Santiago Garrido, Luis Moreno Abstract This paper presents the graphical chromatic representation of the phase and the magnitude

### Prof. D. Manstretta LEZIONI DI FILTRI ANALOGICI. Danilo Manstretta AA

AA-3 LEZIONI DI FILTI ANALOGICI Danilo Manstretta AA -3 AA-3 High Order OA-C Filters H() s a s... a s a s a n s b s b s b s b n n n n... The goal of this lecture is to learn how to design high order OA-C

### Frequency Response. Re ve jφ e jωt ( ) where v is the amplitude and φ is the phase of the sinusoidal signal v(t). ve jφ

27 Frequency Response Before starting, review phasor analysis, Bode plots... Key concept: small-signal models for amplifiers are linear and therefore, cosines and sines are solutions of the linear differential

### MAE 143B - Homework 9

MAE 43B - Homework 9 7.2 2 2 3.8.6.4.2.2 9 8 2 2 3 a) G(s) = (s+)(s+).4.6.8.2.2.4.6.8. Polar plot; red for negative ; no encirclements of, a.s. under unit feedback... 2 2 3. 4 9 2 2 3 h) G(s) = s+ s(s+)..2.4.6.8.2.4