8. Active Filters - 2. Electronic Circuits. Prof. Dr. Qiuting Huang Integrated Systems Laboratory
|
|
- Brianne Andrews
- 6 years ago
- Views:
Transcription
1 8. Active Filters - 2 Electronic Circuits Prof. Dr. Qiuting Huang Integrated Systems Laboratory
2 Blast From The Past: Algebra of Polynomials * PP xx is a polynomial of the variable xx: PP xx = aa 0 + aa 1 xx + + aa nn xx nn Coefficients The highest power of xx in PP xx with non-zero coefficient is called the degree of PP xx. cc 1 is a root of PP xx cc kk is a root of PP xx PP xx = xx cc 1 QQ nn 1 xx. has nn roots PP xx = xx cc 1 xx cc kk QQ nn kk xx. QQ nn kk xx is an irreducible polynomial of degree 1 or 2. has no more roots A real polynomial PP xx can be factorized into degree 1 (linear) and degree 2 (quadratic) polynomials. 2
3 Design of Filters The transfer function of a filter is a rational function. ( ) T s k bks + b s + + bs+ b Ns () = = n as + a s + + as+ a D() s n k 1 k n 1 n The degree of the denominator is usually referred to as the order of the filter (nn). nn is determined by the number of reactive elements in the filter. Example: 0 ( ) = T s Passive 5 th -Order Ladder FIlter b as + as + + as+ a
4 Design of Filters Filter performance improves with the order of the filter. Sharper response with higher nn However, a high-order filter is costly. Optimization process of a filter: Find TT(ss) that satisfies the requirements with lowest possible order. Designing a high-order filter to provide a desired response is not a trivial problem. PB = pass-band TB = transition-band SB = stop-band 4
5 Design of Filters Calibrating the circuit manually is not sufficient to achieve the task. The specifications of the filter are dominated by its worst performer. A systematic mathematical approach is required. Magnitude (db) Chebyshev Low-pass resonse Max. Ripple = 0.1 db Passive 5 th -Order Ladder FIlter The design comprises an optimization process targeting minimizing the maximum deviation from the ideal filter response (minmax optimization). 5
6 Approximation Problem There are several well-known approaches for optimizing the approximation problem stemming from mathematics; i.e. (Butterworth, Chebyshev, Bessel, Elleptic, ). The design parameters and component values for the filter circuits realizing these responses are well-documented. 6
7 Realization Problem: Filter Synthesis We can represent the transfer function obtained from the mathematical approximation as: TT ss = bb kk ss kk + bb kk 1 ss kk bb 1 ss + bb 0 aa nn ss nn + aa nn 1 ss nn aa 1 ss + aa 0 = NN(ss) DD(ss) = bb 1,2 ss 2 + bb 1,1 ss + bb 1,0 bb 2,2 ss 2 + bb 2,1 ss + bb 2,0 bb kk + bb kk 2,2ss2 2,1ss + bb kk 2,0 aa 1,2 ss 2 + aa 1,1 ss + aa 1,0 aa 2,2 ss 2 + aa 2,1 ss + aa 2,0 aann 2,2ss2 + aann 2,1ss + aann 2,0 7
8 Realization Problem: Filter Synthesis We can represent the transfer function obtained from the mathematical approximation as: TT ss = bb kk ss kk + bb kk 1 ss kk bb 1 ss + bb 0 aa nn ss nn + aa nn 1 ss nn aa 1 ss + aa 0 = NN(ss) DD(ss) = KK 1 ss 2 + ωω z1 2 ss + ωω QQ z1 KK 2 ss 2 + ωω z2 2 ss + ωω z1 QQ z2 KKnn z2 2 ss 2 + ωω o1 2 ss + ωω QQ o1 ss 2 + ωω o2 2 ss + ωω o1 QQ o2 ss 2 + o2 Can be implemented as second-order filter biquads ss 2 + ωω o nn 2 QQ o nn 2 ωω z nn 2 QQ z nn 2 2 ss + ωω nn z 2 2 ss + ωω nn o 2 Filter 1 Filter 2 Filter nn 2 8
9 Realization Problem: Filter Synthesis Example: Synthesize a 4 th -order band-pass filter with the normalized transfer function: TT BP ss = 0.01KKss 2 ss ss ss ss+1 Solution: Use 2 nd -order biquads (if the required filter order is odd, add a single 1 st -order section as well). Remember, a BPF can be realized using the cascade of a HPF and a LPF. TT BP = TT HP. TT LP Sallen-Key HPF Sallen-Key LPF TT BP (ss) = KK 1 ss 2 ss ss KK 2 ss ss
10 Realization Problem: Filter Synthesis 2 Alternatively, the synthesis of a transfer function can be achieved by a successive removal of its poles. The removal of a pole corresponds to the extraction of a network element (inductor or capacitor). This process can be accomplished systematically by long division. Example: Synthesize the transfer function TT ss = ss4 +20ss ss 3 +9ss and draw the corresponding circuit diagram. 10
11 Realization Problem: Filter Synthesis 2 Example: Synthesize the transfer function TT ss = ss4 +20ss ss 3 +9ss and draw the corresponding circuit diagram. Solution: Perform long division to obtain: Cauer I Realization: 1 11ss TT ss = ss + 1ss 3 9ss 11ss ss 11ss ss ss 11ss ss + 11 ss ss ss ss 64 ss Ω Series L ss 3 + 9ss ss ss ss 4 + 9ss ss ss 3 + 9ss ss Ω 1 ss Parallel C ss Ω ss ss 11 11ss ss ss 1111 Ω : Inductance Ω 1 : Capacitance ss Ω 1 Series L Parallel C 11
12 Realization Problem: Filter Synthesis 2 Example: Synthesize the transfer function TT ss = ss4 +20ss ss 3 +9ss and draw the corresponding circuit diagram. Solution: Perform long division to obtain: Series L Series L Parallel C Parallel C Cauer I Realization: 1 TT ss = ss ss ss ss 12
13 Passive Filter Design The previous technique is used to systematically synthesize ladder filter structures. Low sensitivity to component mismatch is inherent to ladder filters. LC ladder filter can be singly- or doubly-terminated (with or without RR L ). Double termination minimizes losses and ensures maximum energy transfer. As the filter order nn increases, the quality factor increases. 13
14 From Passive to Active Consider the 5 th -order RLC ladder filter with double termination. Passive 5 th -Order RLC Low-pass Ladder FIlter The Signal Flowgraph (SFG) technique is used to convert the passive ladder circuit into an equivalent active circuit. SFG is a special type of block diagram/directed graph where the variables are represented by nodes, while branches represent the relation between those variables. SFG maps multiplications (weights) & additions (intersections). 14
15 Name currents and voltages for all components: + VV 11 + VV 33 + VV 55 VV 22 VV 44 VV 66 II 11 II 33 II 55 II 22 II 44 II 66 II 77 Derive current and voltage relations using KCL and KVL: VV 1 = VV i VV 2, VV 2 = II 2, VV sscc 3 = VV 2 VV 4, 1 VV 4 = II 4, VV 5 = VV 4 VV 6, VV sscc 6 = II 6, 3 sscc 5 VV o = VV 6, II 1 = VV 1 RR S, II 2 = II 1 II 3, II 3 = VV 3 ssss 2, II 4 = II 3 II 5, II 6 = II 5 II 7, II 5 = VV 5 ssss 4, II 7 = VV 6 RR L. Choose the relations between the voltage and the current. 15
16 Sketch the corresponding SFG using the equations VV 1 = VV i VV 2, VV 2 = II 2, VV sscc 3 = VV 2 VV 4, 1 VV 4 = II 4, VV 5 = VV 4 VV 6, VV sscc 6 = II 6, 3 sscc 5 VV o = VV 6, II 1 = VV 1 RR SS, II 2 = II 1 II 3, II 3 = VV 3 ssss 2, II 4 = II 3 II 5, II 6 = II 5 II 7, II 5 = VV 5 ssss 4, II 7 = VV 6 RR L. 16
17 Re-define the state-variables corresponding to the current nodes at the output of the branches with reactive elements to voltages. 17
18 Convert II 3 into VV 3. 18
19 Convert II 5 into VV 5. 19
20 Re-define the state-variables corresponding to the voltage nodes at the input of the vertical branches to currents. 20
21 Convert VV 1 into II 1. 21
22 Convert VV 3 into II 3. 22
23 Convert VV 5 into II 5. 23
24 Redefine the state variables with negative quantities on their left until there is a single negative branch in each loop. 24
25 Define VV 2A = VV 2. 25
26 Define II 3A = II 3. 26
27 Define VV 3A = VV 3. 27
28 Define II 4A = II 4. 28
29 Define VV 6A = VV 6. 29
30 Define VV oa = VV o. 30
31 Define II 7A = II 7. 31
32 Let the branches containing ss be in the form 1 ssss 32
33 Invert the sign of the positive reactive elements. 33
34 Eliminate the redundant state variables. 34
35 Recall (Lecture 4): The active integrator circuit VV out VV in = 1 ssssss 1 ssττ 35
36 From Passive to Active - Circuit Using the active integrator, we obtain a 5 th order active filter VV o = VV i 36
37 Blast From The Past: Algebra of Polynomials* If FF is a field and nn is a nonnegative integer, then a polynomial PP(xx) of degree nn over FF with aa ii FF for ii = 0,, nn, aa nn 0, and xx an indeterminate, is of the form: PP xx = aa 0 + aa 1 xx + + aa nn xx nn = nn ii=0 aa ii xx ii Denoting the degree of PP(xx) as deg PP xx, let PP xx 0, QQ xx 0 FF. 1. deg PP xx QQ xx = deg(pp xx ) + deg(qq xx ) 2. deg(pp xx ± QQ xx ) max(deg(pp xx ), deg(qq xx )) If ff xx, gg(xx) FF with gg(xx) 0 then gg(xx) divides ff xx, or gg(xx) is a factor of ff(xx), if there exists a polynomial qq(xx) FF such that ff xx = qq xx gg xx. If ff(xx) 0 has no non-trivial, non-unit factors (it cannot be factorized into polynomials of lower degree); then ff(xx) is an irreducible polynomial, or prime polynomial. 37
38 Blast From The Past: Algebra of Polynomials* Hence, there exist unique polynomials qq xx, rr xx FF such that ff xx = qq xx gg xx + rr(xx), where rr xx = 0 or deg rr xx < deg gg xx. The polynomials qq xx and rr xx are called respectively the quotient and remainder. If cc is a root of PP(xx), then (x cc) divides PP(xx), such that PP xx = xx cc QQ(xx) with deg QQ xx = deg PP xx 1. An irreducible polynomial of degree greater than one over a field FF has no roots in FF. Hence, a polynomial of degree nn in FF can have at most nn distinct roots. If PP xx = aa aa nn xx nn is a complex polynomial then its conjugate is the polynomial PP xx = aa aa nn xx nn. That is, the conjugate is the polynomial whose coefficients aa ii are the conjugates of aa 0 of PP(xx). If ff(xx) R and ff zz 0 = 0, then ff zz 0 = 0; the complex roots of real polynomials come in conjugate pairs. 38
39 Blast From The Past: Algebra of Polynomials* Suppose ff(xx) C, with ff(xx) non-constant, then ff(xx) takes on a minimum value at some value zz 0 C, if ff(xx 0 ) 0, then ff(xx 0 ) is not the minimum value of ff(xx) and xx 0 zz 0. Consequently, ff(xx) has at least one complex root. A complex polynomial completely factorizes into linear factors. Additionally, a real polynomial factorizes into degree 1 and degree 2 factors. Equivalently, the only irreducible real polynomials are linear and quadratic polynomials. * Proofs can be found in: Fine, D. and Rosenberger, G. The Fundamental Theorem of Algebra, Springer series on undergraduate texts in Mathematics, Springer,
Math 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationMath 3 Unit 3: Polynomial Functions
Math 3 Unit 3: Polynomial Functions Unit Title Standards 3.1 End Behavior of Polynomial Functions F.IF.7c 3.2 Graphing Polynomial Functions F.IF.7c, A.APR3 3.3 Writing Equations of Polynomial Functions
More informationLesson 1: Successive Differences in Polynomials
Lesson 1 Lesson 1: Successive Differences in Polynomials Classwork Opening Exercise John noticed patterns in the arrangement of numbers in the table below. 2.4 3.4 4.4 5.4 6.4 5.76 11.56 19.36 29.16 40.96
More information9. Switched Capacitor Filters. Electronic Circuits. Prof. Dr. Qiuting Huang Integrated Systems Laboratory
9. Switched Capacitor Filters Electronic Circuits Prof. Dr. Qiuting Huang Integrated Systems Laboratory Motivation Transmission of voice signals requires an active RC low-pass filter with very low ff cutoff
More informationWorksheets for GCSE Mathematics. Quadratics. mr-mathematics.com Maths Resources for Teachers. Algebra
Worksheets for GCSE Mathematics Quadratics mr-mathematics.com Maths Resources for Teachers Algebra Quadratics Worksheets Contents Differentiated Independent Learning Worksheets Solving x + bx + c by factorisation
More informationSections 4.2 and 4.3 Zeros of Polynomial Functions. Complex Numbers
Sections 4.2 and 4.3 Zeros of Polynomial Functions Complex Numbers 1 Sections 4.2 and 4.3 Find the Zeros of Polynomial Functions and Graph Recall from section 4.1 that the end behavior of a polynomial
More informationP.3 Division of Polynomials
00 section P3 P.3 Division of Polynomials In this section we will discuss dividing polynomials. The result of division of polynomials is not always a polynomial. For example, xx + 1 divided by xx becomes
More information10.4 The Cross Product
Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb
More informationThe general form for the transform function of a second order filter is that of a biquadratic (or biquad to the cool kids).
nd-order filters The general form for the transform function of a second order filter is that of a biquadratic (or biquad to the cool kids). T (s) A p s a s a 0 s b s b 0 As before, the poles of the transfer
More information3.4. ZEROS OF POLYNOMIAL FUNCTIONS
3.4. ZEROS OF POLYNOMIAL FUNCTIONS What You Should Learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polynomial functions. Find rational zeros of polynomial functions. Find
More informationTransition to College Math and Statistics
Transition to College Math and Statistics Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear College Algebra Students, This assignment
More informationMATH 1080: Calculus of One Variable II Fall 2018 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of One Variable II Fall 2018 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart Unit 2 Skill Set Important: Students should expect test questions that require
More informationAnalog Circuits and Systems
Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 27: State Space Filters 1 Review Q enhancement of passive RC using negative and positive feedback Effect of finite GB of the active device on
More informationElectronic Circuits. Prof. Dr. Qiuting Huang Integrated Systems Laboratory
Electronic Circuits Prof. Dr. Qiuting Huang 6. Transimpedance Amplifiers, Voltage Regulators, Logarithmic Amplifiers, Anti-Logarithmic Amplifiers Transimpedance Amplifiers Sensing an input current ii in
More informationSECTION 5: POWER FLOW. ESE 470 Energy Distribution Systems
SECTION 5: POWER FLOW ESE 470 Energy Distribution Systems 2 Introduction Nodal Analysis 3 Consider the following circuit Three voltage sources VV sss, VV sss, VV sss Generic branch impedances Could be
More informationOPERATIONAL 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
More informationProf. 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
More informationLogarithmic Functions
Name Student ID Number Group Name Group Members Logarithmic Functions 1. Solve the equations below. xx = xx = 5. Were you able solve both equations above? If so, was one of the equations easier to solve
More informationOp-Amp Circuits: Part 3
Op-Amp Circuits: Part 3 M. B. Patil mbpatil@ee.iitb.ac.in www.ee.iitb.ac.in/~sequel Department of Electrical Engineering Indian Institute of Technology Bombay Introduction to filters Consider v(t) = v
More informationECE3050 Assignment 7
ECE3050 Assignment 7. Sketch and label the Bode magnitude and phase plots for the transfer functions given. Use loglog scales for the magnitude plots and linear-log scales for the phase plots. On the magnitude
More informationSophomore Physics Laboratory (PH005/105)
CALIFORNIA INSTITUTE OF TECHNOLOGY PHYSICS MATHEMATICS AND ASTRONOMY DIVISION Sophomore Physics Laboratory (PH5/15) Analog Electronics Active Filters Copyright c Virgínio de Oliveira Sannibale, 23 (Revision
More informationFilters and Tuned Amplifiers
Filters and Tuned Amplifiers Essential building block in many systems, particularly in communication and instrumentation systems Typically implemented in one of three technologies: passive LC filters,
More informationSecond-order filters. EE 230 second-order filters 1
Second-order filters Second order filters: Have second order polynomials in the denominator of the transfer function, and can have zeroth-, first-, or second-order polynomials in the numerator. Use two
More informationQuadratic Equations and Functions
50 Quadratic Equations and Functions In this chapter, we discuss various ways of solving quadratic equations, aaxx 2 + bbbb + cc 0, including equations quadratic in form, such as xx 2 + xx 1 20 0, and
More informationTEXT AND OTHER MATERIALS:
1. TEXT AND OTHER MATERIALS: Check Learning Resources in shared class files Calculus Wiki-book: https://en.wikibooks.org/wiki/calculus (Main Reference e-book) Paul s Online Math Notes: http://tutorial.math.lamar.edu
More informationdue date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish)
Honors PreCalculus Summer Work 016 due date: third day of class estimated time: 10 hours (for planning purposes only; work until you finish) Dear Honors PreCalculus Students, This assignment is designed
More information2nd-order filters. EE 230 second-order filters 1
nd-order filters Second order filters: Have second order polynomials in the denominator of the transfer function, and can have zeroth-, first-, or second-order polyinomials in the numerator. Use two reactive
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
MCA AP Calculus AB Summer Assignment The following packet is a review of many of the skills needed as we begin the study of Calculus. There two major sections to this review. Pages 2-9 are review examples
More informationExam Programme VWO Mathematics A
Exam Programme VWO Mathematics A The exam. The exam programme recognizes the following domains: Domain A Domain B Domain C Domain D Domain E Mathematical skills Algebra and systematic counting Relationships
More informationPolynomials and Polynomial Functions
1 Polynomials and Polynomial Functions One of the simplest types of algebraic expressions are polynomials. They are formed only by addition and multiplication of variables and constants. Since both addition
More informationELEC273 Lecture Notes Set 11 AC Circuit Theorems
ELEC273 Lecture Notes Set C Circuit Theorems The course web site is: http://users.encs.concordia.ca/~trueman/web_page_273.htm Final Exam (confirmed): Friday December 5, 207 from 9:00 to 2:00 (confirmed)
More informationP.2 Multiplication of Polynomials
1 P.2 Multiplication of Polynomials aa + bb aa + bb As shown in the previous section, addition and subtraction of polynomials results in another polynomial. This means that the set of polynomials is closed
More informationUnit 6 Note Packet List of topics for this unit/assignment tracker Date Topic Assignment & Due Date Absolute Value Transformations Day 1
Name: Period: Unit 6 Note Packet List of topics for this unit/assignment tracker Date Topic Assignment & Due Date Absolute Value Transformations Day 1 Absolute Value Transformations Day 2 Graphing Equations
More informationIntegration of Rational Functions by Partial Fractions
Integration of Rational Functions by Partial Fractions Part 2: Integrating Rational Functions Rational Functions Recall that a rational function is the quotient of two polynomials. x + 3 x + 2 x + 2 x
More informationIntegrating Rational functions by the Method of Partial fraction Decomposition. Antony L. Foster
Integrating Rational functions by the Method of Partial fraction Decomposition By Antony L. Foster At times, especially in calculus, it is necessary, it is necessary to express a fraction as the sum of
More informationSpeaker: Arthur Williams Chief Scientist Telebyte Inc. Thursday November 20 th 2008 INTRODUCTION TO ACTIVE AND PASSIVE ANALOG
INTRODUCTION TO ACTIVE AND PASSIVE ANALOG FILTER DESIGN INCLUDING SOME INTERESTING AND UNIQUE CONFIGURATIONS Speaker: Arthur Williams Chief Scientist Telebyte Inc. Thursday November 20 th 2008 TOPICS Introduction
More information3.2 A2 - Just Like Derivatives but Backwards
3. A - Just Like Derivatives but Backwards The Definite Integral In the previous lesson, you saw that as the number of rectangles got larger and larger, the values of Ln, Mn, and Rn all grew closer and
More information3. Mathematical Modelling
3. Mathematical Modelling 3.1 Modelling principles 3.1.1 Model types 3.1.2 Model construction 3.1.3 Modelling from first principles 3.2 Models for technical systems 3.2.1 Electrical systems 3.2.2 Mechanical
More informationTo find the step response of an RC circuit
To find the step response of an RC circuit v( t) v( ) [ v( t) v( )] e tt The time constant = RC The final capacitor voltage v() The initial capacitor voltage v(t ) To find the step response of an RL circuit
More informationDeliyannis, Theodore L. et al "Two Integrator Loop OTA-C Filters" Continuous-Time Active Filter Design Boca Raton: CRC Press LLC,1999
Deliyannis, Theodore L. et al "Two Integrator Loop OTA-C Filters" Continuous-Time Active Filter Design Boca Raton: CRC Press LLC,1999 Chapter 9 Two Integrator Loop OTA-C Filters 9.1 Introduction As discussed
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationCase Study: Parallel Coupled- Line Combline Filter
MICROWAVE AND RF DESIGN MICROWAVE AND RF DESIGN Case Study: Parallel Coupled- Line Combline Filter Presented by Michael Steer Reading: 6. 6.4 Index: CS_PCL_Filter Based on material in Microwave and RF
More informationRational Expressions and Functions
Rational Expressions and Functions In the previous two chapters we discussed algebraic expressions, equations, and functions related to polynomials. In this chapter, we will examine a broader category
More informationLecture 3. Logic Predicates and Quantified Statements Statements with Multiple Quantifiers. Introduction to Proofs. Reading (Epp s textbook)
Lecture 3 Logic Predicates and Quantified Statements Statements with Multiple Quantifiers Reading (Epp s textbook) 3.1-3.3 Introduction to Proofs Reading (Epp s textbook) 4.1-4.2 1 Propositional Functions
More informationAC 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
More informationMaster Degree in Electronic Engineering. Analog and Telecommunication Electronics course Prof. Del Corso Dante A.Y Switched Capacitor
Master Degree in Electronic Engineering TOP-UIC Torino-Chicago Double Degree Project Analog and Telecommunication Electronics course Prof. Del Corso Dante A.Y. 2013-2014 Switched Capacitor Working Principles
More informationMath 3 Unit 4: Rational Functions
Math Unit : Rational Functions Unit Title Standards. Equivalent Rational Expressions A.APR.6. Multiplying and Dividing Rational Expressions A.APR.7. Adding and Subtracting Rational Expressions A.APR.7.
More informationLecture No. 1 Introduction to Method of Weighted Residuals. Solve the differential equation L (u) = p(x) in V where L is a differential operator
Lecture No. 1 Introduction to Method of Weighted Residuals Solve the differential equation L (u) = p(x) in V where L is a differential operator with boundary conditions S(u) = g(x) on Γ where S is a differential
More informationEE 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)
More informationProcedure for Graphing Polynomial Functions
Procedure for Graphing Polynomial Functions P(x) = a nx n + a n-1x n-1 + + a 1x + a 0 To graph P(x): As an example, we will examine the following polynomial function: P(x) = 2x 3 3x 2 23x + 12 1. Determine
More informationAdjoint networks and other elements of circuit theory. E416 4.Adjoint networks
djoint networks and other elements of circuit theory One-port reciprocal networks one-port network is reciprocal if: V I I V = Where and are two different tests on the element Example: a linear impedance
More informationMathematical Induction Assignments
1 Mathematical Induction Assignments Prove the Following using Principle of Mathematical induction 1) Prove that for any positive integer number n, n 3 + 2 n is divisible by 3 2) Prove that 1 3 + 2 3 +
More informationLesson 24: Using the Quadratic Formula,
, b ± b 4ac x = a Opening Exercise 1. Examine the two equation below and discuss what is the most efficient way to solve each one. A. 4xx + 5xx + 3 = xx 3xx B. cc 14 = 5cc. Solve each equation with the
More informationF.1 Greatest Common Factor and Factoring by Grouping
section F1 214 is the reverse process of multiplication. polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example,
More informationLaplace 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.
More informationPolynomials and Polynomial Functions
1 Polynomials and Polynomial Functions One of the simplest types of algebraic expressions are polynomials. They are formed only by addition and multiplication of variables and constants. Since both addition
More informationSingle-Time-Constant (STC) Circuits This lecture is given as a background that will be needed to determine the frequency response of the amplifiers.
Single-Time-Constant (STC) Circuits This lecture is given as a background that will be needed to determine the frequency response of the amplifiers. Objectives To analyze and understand STC circuits with
More informationEE221 Circuits II. Chapter 14 Frequency Response
EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science : Circuits & Electronics Problem Set #1 Solution
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.2: Circuits & Electronics Problem Set # Solution Exercise. The three resistors form a series connection.
More informationEE221 Circuits II. Chapter 14 Frequency Response
EE22 Circuits II Chapter 4 Frequency Response Frequency Response Chapter 4 4. Introduction 4.2 Transfer Function 4.3 Bode Plots 4.4 Series Resonance 4.5 Parallel Resonance 4.6 Passive Filters 4.7 Active
More informationTotal No. of Questions :09] [Total No. of Pages : 03
EE 4 (RR) Total No. of Questions :09] [Total No. of Pages : 03 II/IV B.Tech. DEGREE EXAMINATIONS, APRIL/MAY- 016 Second Semester ELECTRICAL & ELECTRONICS NETWORK ANALYSIS Time: Three Hours Answer Question
More informationMath 171 Spring 2017 Final Exam. Problem Worth
Math 171 Spring 2017 Final Exam Problem 1 2 3 4 5 6 7 8 9 10 11 Worth 9 6 6 5 9 8 5 8 8 8 10 12 13 14 15 16 17 18 19 20 21 22 Total 8 5 5 6 6 8 6 6 6 6 6 150 Last Name: First Name: Student ID: Section:
More informationSection 3.6 Complex Zeros
04 Chapter Section 6 Complex Zeros When finding the zeros of polynomials, at some point you're faced with the problem x = While there are clearly no real numbers that are solutions to this equation, leaving
More informationNetwork Synthesis. References :
References : Network ynthesis Gabor C. Temes & Jack W. Lapatra, Introduction to Circuit ynthesis and Design, McGraw-Hill Book Company. M.E. Van Valkenburg, Introduction to Modern Network ynthesis, John
More information10.1 Three Dimensional Space
Math 172 Chapter 10A notes Page 1 of 12 10.1 Three Dimensional Space 2D space 0 xx.. xx-, 0 yy yy-, PP(xx, yy) [Fig. 1] Point PP represented by (xx, yy), an ordered pair of real nos. Set of all ordered
More informationMathematical Methods 2019 v1.2
Problem-solving and modelling task (20%) This sample has been compiled by the QCAA to assist and support teachers to match evidence in student responses to the characteristics described in the instrument-specific
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 05 IIR Design 14/03/04 http://www.ee.unlv.edu/~b1morris/ee482/
More informationNetwork 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
More informationZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS
ZEROS OF POLYNOMIAL FUNCTIONS ALL I HAVE TO KNOW ABOUT POLYNOMIAL FUNCTIONS TOOLS IN FINDING ZEROS OF POLYNOMIAL FUNCTIONS Synthetic Division and Remainder Theorem (Compressed Synthetic Division) Fundamental
More informationF.3 Special Factoring and a General Strategy of Factoring
F.3 Special Factoring and a General Strategy of Factoring Difference of Squares section F4 233 Recall that in Section P2, we considered formulas that provide a shortcut for finding special products, such
More informationEE 40: Introduction to Microelectronic Circuits Spring 2008: Midterm 2
EE 4: Introduction to Microelectronic Circuits Spring 8: Midterm Venkat Anantharam 3/9/8 Total Time Allotted : min Total Points:. This is a closed book exam. However, you are allowed to bring two pages
More informationSystems of Linear Equations
Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has
More informationElectronic Circuits Summary
Electronic Circuits Summary Andreas Biri, D-ITET 6.06.4 Constants (@300K) ε 0 = 8.854 0 F m m 0 = 9. 0 3 kg k =.38 0 3 J K = 8.67 0 5 ev/k kt q = 0.059 V, q kt = 38.6, kt = 5.9 mev V Small Signal Equivalent
More informationRadicals and Radical Functions
0 Radicals and Radical Functions So far we have discussed polynomial and rational expressions and functions. In this chapter, we study algebraic expressions that contain radicals. For example, + 2, xx,
More informationSome of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e
Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)
More informationF.4 Solving Polynomial Equations and Applications of Factoring
section F4 243 F.4 Zero-Product Property Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or first-degree, equations. Solving higher
More informationx 2 + 6x 18 x + 2 Name: Class: Date: 1. Find the coordinates of the local extreme of the function y = x 2 4 x.
1. Find the coordinates of the local extreme of the function y = x 2 4 x. 2. How many local maxima and minima does the polynomial y = 8 x 2 + 7 x + 7 have? 3. How many local maxima and minima does the
More informationSect Introduction to Rational Expressions
127 Sect 7.1 - Introduction to Rational Expressions Concept #1 Definition of a Rational Expression. Recall that a rational number is any number that can be written as the ratio of two integers where the
More informationEE40 Midterm Review Prof. Nathan Cheung
EE40 Midterm Review Prof. Nathan Cheung 10/29/2009 Slide 1 I feel I know the topics but I cannot solve the problems Now what? Slide 2 R L C Properties Slide 3 Ideal Voltage Source *Current depends d on
More informationEE 508 Lecture 22. Sensitivity Functions - Comparison of Circuits - Predistortion and Calibration
EE 58 Lecture Sensitivity Functions - Comparison of Circuits - Predistortion and Calibration Review from last time Sensitivity Comparisons Consider 5 second-order lowpass filters (all can realize same
More informationF.1 Greatest Common Factor and Factoring by Grouping
1 Factoring Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers.
More informationUNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS
UNIT 4 DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS 1.0 Kirchoff s Law Kirchoff s Current Law (KCL) states at any junction in an electric circuit the total current flowing towards that junction is equal
More informationH(s) = 2(s+10)(s+100) (s+1)(s+1000)
Problem 1 Consider the following transfer function H(s) = 2(s10)(s100) (s1)(s1000) (a) Draw the asymptotic magnitude Bode plot for H(s). Solution: The transfer function is not in standard form to sketch
More informationDay 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5
Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There
More information2.4 Error Analysis for Iterative Methods
2.4 Error Analysis for Iterative Methods 1 Definition 2.7. Order of Convergence Suppose {pp nn } nn=0 is a sequence that converges to pp with pp nn pp for all nn. If positive constants λλ and αα exist
More informationClassical RSA algorithm
Classical RSA algorithm We need to discuss some mathematics (number theory) first Modulo-NN arithmetic (modular arithmetic, clock arithmetic) 9 (mod 7) 4 3 5 (mod 7) congruent (I will also use = instead
More informationA. Incorrect! Apply the rational root test to determine if any rational roots exist.
College Algebra - Problem Drill 13: Zeros of Polynomial Functions No. 1 of 10 1. Determine which statement is true given f() = 3 + 4. A. f() is irreducible. B. f() has no real roots. C. There is a root
More informationDigital Control & Digital Filters. Lectures 21 & 22
Digital Controls & Digital Filters Lectures 2 & 22, Professor Department of Electrical and Computer Engineering Colorado State University Spring 205 Review of Analog Filters-Cont. Types of Analog Filters:
More informationSupport Vector Machines. CSE 4309 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington
Support Vector Machines CSE 4309 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1 A Linearly Separable Problem Consider the binary classification
More informationPartial Fractions. Prerequisites: Solving simple equations; comparing coefficients; factorising simple quadratics and cubics; polynomial division.
Prerequisites: olving simple equations; comparing coefficients; factorising simple quadratics and cubics; polynomial division. Maths Applications: Integration; graph sketching. Real-World Applications:
More informationLogic and Proofs. (A brief summary)
Logic and Proofs (A brief summary) Why Study Logic: To learn to prove claims/statements rigorously To be able to judge better the soundness and consistency of (others ) arguments To gain the foundations
More informationSECTION 8: ROOT-LOCUS ANALYSIS. ESE 499 Feedback Control Systems
SECTION 8: ROOT-LOCUS ANALYSIS ESE 499 Feedback Control Systems 2 Introduction Introduction 3 Consider a general feedback system: Closed-loop transfer function is KKKK ss TT ss = 1 + KKKK ss HH ss GG ss
More informationResponse of Second-Order Systems
Unit 3 Response of SecondOrder Systems In this unit, we consider the natural and step responses of simple series and parallel circuits containing inductors, capacitors and resistors. The equations which
More informationSpecialist Mathematics 2019 v1.2
Examination (15%) This sample has been compiled by the QCAA to assist and support teachers in planning and developing assessment instruments for individual school settings. The examination must ensure
More informationEE-202 Exam III April 13, 2015
EE-202 Exam III April 3, 205 Name: (Please print clearly.) Student ID: CIRCLE YOUR DIVISION DeCarlo-7:30-8:30 Furgason 3:30-4:30 DeCarlo-:30-2:30 202 2022 2023 INSTRUCTIONS There are 2 multiple choice
More informationSinusoidal Steady-State Analysis
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.
More informationThe standard form for a general polynomial of degree n is written. Examples of a polynomial in standard form
Section 4 1A: The Rational Zeros (Roots) of a Polynomial The standard form for a general polynomial of degree n is written f (x) = a n x n + a n 1 x n 1 +... + a 1 x + a 0 where the highest degree term
More informationSECTION 7: STEADY-STATE ERROR. ESE 499 Feedback Control Systems
SECTION 7: STEADY-STATE ERROR ESE 499 Feedback Control Systems 2 Introduction Steady-State Error Introduction 3 Consider a simple unity-feedback system The error is the difference between the reference
More informationQUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34)
QUESTION BANK SUBJECT: NETWORK ANALYSIS (10ES34) NOTE: FOR NUMERICAL PROBLEMS FOR ALL UNITS EXCEPT UNIT 5 REFER THE E-BOOK ENGINEERING CIRCUIT ANALYSIS, 7 th EDITION HAYT AND KIMMERLY. PAGE NUMBERS OF
More informationSECTION 7: FAULT ANALYSIS. ESE 470 Energy Distribution Systems
SECTION 7: FAULT ANALYSIS ESE 470 Energy Distribution Systems 2 Introduction Power System Faults 3 Faults in three-phase power systems are short circuits Line-to-ground Line-to-line Result in the flow
More information