Stepped-Impedance Low-Pass Filters
|
|
- Conrad Terence French
- 6 years ago
- Views:
Transcription
1 4/23/27 Stepped Impedance Low Pass Filters 1/14 Stepped-Impedance Low-Pass Filters Say we know te impedance matrix of a symmetric two-port device: = Regardless of te construction of tis two port device, we can model it as a simple T-circuit, consisting of tree impedances: Port 1 Port In oter words, if te two series impedances ave an impedance value equal to, and te sunt impedance as a value equal to, te impedance matrix of tis Tcircuit is: = Tus, any symmetric two-port network can be modeled by tis T-circuit! Jim Stiles Te Univ. of Kansas Dept. of EECS
2 4/23/27 Stepped Impedance Low Pass Filters 2/14 For example, consider a lengt of transmission line (a symmetric two-port network!): Recall (or determine for yourself!) tat te impedance parameters of tis two port network are: = = j cot β = = j csc β Wit a little trigonometry, ICBST : j tan β = 2 Furtermore, if β is small: sin β β cos β 1 tan β β were β is expressed in radians. Tus, j β 2 and also: Jim Stiles Te Univ. of Kansas Dept. of EECS
3 4/23/27 Stepped Impedance Low Pass Filters 3/14 = = j β csc j β Tus, an electrically sort ( β 1) transmission line can be approximately modeled wit a T-circuit as: j β 2 j β 2 Port 1 j β Port 2 Now, consider also te case were te caracteristic impedance of te transmission line is relatively large. We ll denote tis large caracteristic impedance as. Note te sunt impedance, value j β. Since te numerator ( ) is relatively large, and te denominator ( j β ) is small, te impedance sunt device is very large. So large, in fact, tat we can approximate it as an open circuit! for β1 and j β So now we ave a furter simplification of our model: Jim Stiles Te Univ. of Kansas Dept. of EECS
4 4/23/27 Stepped Impedance Low Pass Filters 4/14 j β 2 j β 2 Port 1 Port 2 Te remaining impedances are now in series, so te circuit can be furter simplified to: j β Port 1 Port 2 Te equivalent circuit for transmission line wit sort electrical lengt ( β 1) and large caracteristic impedance ( ). Now, consider te case were te caracteristic impedance of te transmission line as a relatively low value, denoted as, were. Note te series impedance, values j ( β ) 2. Since bot and β are small, te product of te two is very small. So small, in fact, tat we can approximate it as a sort circuit! Jim Stiles Te Univ. of Kansas Dept. of EECS
5 4/23/27 Stepped Impedance Low Pass Filters 5/14 β j for β1 and 2 So now we ave anoter simplification of our model: Port 1 j β Port 2 Wic of course furter simplifies to: Port 1 j β Port 2 Te equivalent circuit for transmission line wit sort electrical lengt ( β 1) and small caracteristic impedance ( ). Q: But, wat does all tis ave to do wit constructing a low-pass filter? A: Plenty! Recall tat a lossless low-pass filter constructed wit lumped elements consists of a circuit ladder of series inductors and sunt capacitors! Jim Stiles Te Univ. of Kansas Dept. of EECS
6 4/23/27 Stepped Impedance Low Pass Filters 6/14 Q: So? A: Look at te two equivalent circuits for an electrically sort transmission line. Te one wit large caracteristic impedance as te form of a series inductor, and te one wit small caracteristic impedance as te form of a sunt capacitor! I.E.: jx = j β and: j X = j ωl are identical if: j β = jωl β = ωl Tus, te series inductance of our transmission line lengt is: Jim Stiles Te Univ. of Kansas Dept. of EECS
7 4/23/27 Stepped Impedance Low Pass Filters 7/14 L = ω β Q: Yikes! Our inductance appears to be a function of frequency ω. I assume we simply set tis value to cutoff frequency ω, just like we did for Ricard s transformation? c A: Nope! We can simplify te result a bit more. Recall tat β = ω v p, so tat: β L = = ω v p In oter words, te series impedance resulting from our sort transmission line is: jω = v p Q: Wow! Tis realization seems to give us a result tat precisely matces an inductor at all frequencies rigt? A: Not quite! Recall tis result was obtained from applying a few approximations te result is not exact! Moreover, one of tese approximations was tat te electrical lengt of te transmission line be small. Tis obviously cannot be true at all frequencies. As te signal frequency increases, so does te electrical lengt our approximate solution will no longer be valid. Jim Stiles Te Univ. of Kansas Dept. of EECS
8 4/23/27 Stepped Impedance Low Pass Filters 8/14 us, tis realization is accurate only for low frequencies recall tat was likewise true for Ricard s transformations! Q: Low-frequencies? How low is low? A: Well, for our filter to provide a response tat accurately follows te lumped element design, our approximation sould be valid for frequencies up to (and including!) te filter cutoff frequency ω c. A general rule-of-tumb is tat a small electrical lengt is defined as being less tan π 4 radians. Tus, to maintain tis small electrical lengt at frequency ω c, our realization must satisfy te relationsip: ω L π β c = < c 4 Note tat tis criterion is difficult to satisfy if te filter cutoff frequency and/or te inductance value L tat we are trying to realize is large. Our only recourse for tese callenging conditions is to increase te value of caracteristic impedance. Q: Is tere some particular difficulty wit increasing? Jim Stiles Te Univ. of Kansas Dept. of EECS
9 4/23/27 Stepped Impedance Low Pass Filters 9/14 A: Could be! Tere is always a practical limit to ow large (or small) we can make te caracteristic impedance of a transmission line. For example, a large caracteristic impedance implemented in microstip/stripline requires a very narrow conductor widt W. But manufacturing tolerances, power andling capability and/or line loss (line resistance R increases as W decreases) place a lower bound on ow narrow we can make tese conductors! However, assuming tat we can satisfy te above constraint, we can approximately realize a lumped inductor of inductance value L by selecting te correct caracteristic impedance and line lengt of our sort transmission line: L = v p Q: For Ricard s Transformation, we first set te stub lengt to a fixed value (i.e., = λ c 8), and ten determined te specific caracteristic impedance necessary to realize a specific inductor value L. I assume we follow te same procedure ere? A: Nope! Wen constructing stepped-impedance low-pass filters, we typically do te opposite! Jim Stiles Te Univ. of Kansas Dept. of EECS
10 4/23/27 Stepped Impedance Low Pass Filters 1/14 1) First, we select te value of, making sure tat te sort electrical lengt inequality is satisfied for te largest inductance value L in our lumped element filter: > 4 ω π c L Tis caracteristic impedance value is typically used to realize all inductor values L in our low-pass filter, regardless of te actual value of inductance L. 2) Ten, we determine te specific lengts n of te transmission line required to realize specific filter inductors values L n : n v p = L n Q: Wat about te sunt capacitors? A: Almost forgot! Recall te low-impedance transmission line provided a sunt impedance tat matced a sunt capacitor: Jim Stiles Te Univ. of Kansas Dept. of EECS
11 4/23/27 Stepped Impedance Low Pass Filters 11/14 I.E.: j β and: j ωc are identical if: j j ωc β ωc = 1 β = Tus, te sunt capacitance of our transmission line lengt is: β C = ω But again using te fact tat β = ω v : p C = v p Jim Stiles Te Univ. of Kansas Dept. of EECS
12 4/23/27 Stepped Impedance Low Pass Filters 12/14 And tus te sunt reactance of our transmission line realization is: v p j = ω Altoug tis again appears to provide exactly te same beavior as a capacitor (as a function of frequency), it is likewise accurate only for low frequencies, were β < π 4. Tus from our realization equality: β = ωc We can conclude tat for our approximations to be valid at all frequencies up to te filter cutoff frequency, te following inequality must be valid: β π = ω C < 4 c c Note tat for difficult design cases were ω c and/or C is very large, te line caracteristic impedance must be made very small. Q: I suppose tere is likewise a problem wit making very small? Jim Stiles Te Univ. of Kansas Dept. of EECS
13 4/23/27 Stepped Impedance Low Pass Filters 13/14 A: Yes! In microstrip and stripline, making small means making conductor widt W very large. In oter words, it will take up lots of space on our substrate. For most applications te surface area of te substrate is bot limited and precious, and tus tere is generally a practical limit on ow wide we can make widt W (i.e., ow low we can make ). However, assuming tat we can satisfy te above constraint, we can approximately realize a lumped capacitor of inductance value C by selecting te correct caracteristic impedance and line lengt of our sort transmission line: C = v p Te design rules for sunt capacitor realization are tus: 1) First, we select te value of, making sure tat te sort electrical lengt inequality is satisfied for te largest capacitance value C in our lumped element filter: < π 4ω c C Jim Stiles Te Univ. of Kansas Dept. of EECS
14 4/23/27 Stepped Impedance Low Pass Filters 14/14 Tis caracteristic impedance value is typically used to realize all capacitor values C in our low-pass filter, regardless of te actual value of capacitance C. 2) Ten, we determine te specific lengts n of te transmission line required to realize specific filter capacitor values C n : = ( v ) C n p n An example of a low-pass, stepped-impedance filter design is provided on page of your book (but of course, you already knew tat rigt?). Jim Stiles Te Univ. of Kansas Dept. of EECS
Continuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;
More informationIntroduction to Derivatives
Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))
More informationExam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
More informationFunction Composition and Chain Rules
Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function
More information1. Which one of the following expressions is not equal to all the others? 1 C. 1 D. 25x. 2. Simplify this expression as much as possible.
004 Algebra Pretest answers and scoring Part A. Multiple coice questions. Directions: Circle te letter ( A, B, C, D, or E ) net to te correct answer. points eac, no partial credit. Wic one of te following
More information3.4 Worksheet: Proof of the Chain Rule NAME
Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are
More information1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
More informationMath 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
More informationMicrostrip Antennas- Rectangular Patch
April 4, 7 rect_patc_tl.doc Page of 6 Microstrip Antennas- Rectangular Patc (Capter 4 in Antenna Teory, Analysis and Design (nd Edition) by Balanis) Sown in Figures 4. - 4.3 Easy to analyze using transmission
More information5.1 We will begin this section with the definition of a rational expression. We
Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go
More informationDerivatives of Exponentials
mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function
More informationCombining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
More informationAN IMPROVED WEIGHTED TOTAL HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES
AN IMPROVED WEIGHTED TOTA HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES Tomas A. IPO University of Wisconsin, 45 Engineering Drive, Madison WI, USA P: -(608)-6-087, Fax: -(608)-6-5559, lipo@engr.wisc.edu
More information1watt=1W=1kg m 2 /s 3
Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory
More information1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2
MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are
More information1 + t5 dt with respect to x. du = 2. dg du = f(u). du dx. dg dx = dg. du du. dg du. dx = 4x3. - page 1 -
Eercise. Find te derivative of g( 3 + t5 dt wit respect to. Solution: Te integrand is f(t + t 5. By FTC, f( + 5. Eercise. Find te derivative of e t2 dt wit respect to. Solution: Te integrand is f(t e t2.
More informationPre-Calculus Review Preemptive Strike
Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly
More informationTHE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
More informationf a h f a h h lim lim
Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point
More informationPreface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser
More information4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
More information3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
More informationHOMEWORK HELP 2 FOR MATH 151
HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,
More informationOn my honor as a student, I have neither given nor received unauthorized assistance on this exam.
HW2 (Overview of Transport) (Print name above) On my onor as a student, I ave neiter given nor received unautorized assistance on tis exam. (sign name above) 1 Figure 1: Band-diagram before and after application
More informationContinuity and Differentiability of the Trigonometric Functions
[Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te
More informationLecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
More informationNumerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
More informationNUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
More information2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
More informationMath 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006
Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2
More informationWYSE Academic Challenge 2004 Sectional Mathematics Solution Set
WYSE Academic Callenge 00 Sectional Matematics Solution Set. Answer: B. Since te equation can be written in te form x + y, we ave a major 5 semi-axis of lengt 5 and minor semi-axis of lengt. Tis means
More informationSection 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.
Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope
More informationConsider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.
Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions
More information2.3 Algebraic approach to limits
CHAPTER 2. LIMITS 32 2.3 Algebraic approac to its Now we start to learn ow to find its algebraically. Tis starts wit te simplest possible its, and ten builds tese up to more complicated examples. Fact.
More informationMaterial for Difference Quotient
Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient
More informationQuaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 01 Aug 08, 2016.
Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals Gary D. Simpson gsim1887@aol.com rev 1 Aug 8, 216 Summary Definitions are presented for "quaternion functions" of a quaternion. Polynomial
More informationPrecalculus Test 2 Practice Questions Page 1. Note: You can expect other types of questions on the test than the ones presented here!
Precalculus Test 2 Practice Questions Page Note: You can expect oter types of questions on te test tan te ones presented ere! Questions Example. Find te vertex of te quadratic f(x) = 4x 2 x. Example 2.
More informationLIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION
LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y
More informationExcerpt from "Calculus" 2013 AoPS Inc.
Excerpt from "Calculus" 03 AoPS Inc. Te term related rates refers to two quantities tat are dependent on eac oter and tat are canging over time. We can use te dependent relationsip between te quantities
More informationDC and AC Impedance of Reactive Elements
3/6/20 D and A Impedance of Reactive Elements /6 D and A Impedance of Reactive Elements Now, recall from EES 2 the complex impedances of our basic circuit elements: ZR = R Z = jω ZL = jωl For a D signal
More informationINTRODUCTION AND MATHEMATICAL CONCEPTS
Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips
More informationSolution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.
December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need
More informationA.P. CALCULUS (AB) Outline Chapter 3 (Derivatives)
A.P. CALCULUS (AB) Outline Capter 3 (Derivatives) NAME Date Previously in Capter 2 we determined te slope of a tangent line to a curve at a point as te limit of te slopes of secant lines using tat point
More informationHOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS
HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS Po-Ceng Cang National Standard Time & Frequency Lab., TL, Taiwan 1, Lane 551, Min-Tsu Road, Sec. 5, Yang-Mei, Taoyuan, Taiwan 36 Tel: 886 3
More informationMath 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
More informationHigher Derivatives. Differentiable Functions
Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.
More informationMath 1210 Midterm 1 January 31st, 2014
Mat 110 Midterm 1 January 1st, 01 Tis exam consists of sections, A and B. Section A is conceptual, wereas section B is more computational. Te value of every question is indicated at te beginning of it.
More information(a) At what number x = a does f have a removable discontinuity? What value f(a) should be assigned to f at x = a in order to make f continuous at a?
Solutions to Test 1 Fall 016 1pt 1. Te grap of a function f(x) is sown at rigt below. Part I. State te value of eac limit. If a limit is infinite, state weter it is or. If a limit does not exist (but is
More informationThe Derivative The rate of change
Calculus Lia Vas Te Derivative Te rate of cange Knowing and understanding te concept of derivative will enable you to answer te following questions. Let us consider a quantity wose size is described by
More informationKrazy Katt, the mechanical cat
Krazy Katt, te mecanical cat Te cat rigting relex is a cat's innate ability to orient itsel as it alls in order to land on its eet. Te rigting relex begins to appear at 3 4 weeks o age, and is perected
More informationLecture 12 Date:
Lecture 12 Date: 09.02.2017 Microstrip Matching Networks Series- and Shunt-stub Matching Quarter Wave Impedance Transformer Microstrip Line Matching Networks In the lower RF region, its often a standard
More information. Compute the following limits.
Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim
More informationLesson 6: The Derivative
Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange
More informationHow to measure complex impedance at high frequencies where phase measurement is unreliable.
Objectives In this course you will learn the following Various applications of transmission lines. How to measure complex impedance at high frequencies where phase measurement is unreliable. How and why
More informationSin, Cos and All That
Sin, Cos and All Tat James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 9, 2017 Outline Sin, Cos and all tat! A New Power Rule Derivatives
More informationReview for Exam IV MATH 1113 sections 51 & 52 Fall 2018
Review for Exam IV MATH 111 sections 51 & 52 Fall 2018 Sections Covered: 6., 6., 6.5, 6.6, 7., 7.1, 7.2, 7., 7.5 Calculator Policy: Calculator use may be allowed on part of te exam. Wen instructions call
More informationFunction Composition and Chain Rules
Function Composition an Cain Rules James K. Peterson Department of Biological Sciences an Department of Matematical Sciences Clemson University November 2, 2018 Outline Function Composition an Continuity
More informationBob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk
Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of
More informationCopyright c 2008 Kevin Long
Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula
More informationExponentials and Logarithms Review Part 2: Exponentials
Eponentials and Logaritms Review Part : Eponentials Notice te difference etween te functions: g( ) and f ( ) In te function g( ), te variale is te ase and te eponent is a constant. Tis is called a power
More informationMath 161 (33) - Final exam
Name: Id #: Mat 161 (33) - Final exam Fall Quarter 2015 Wednesday December 9, 2015-10:30am to 12:30am Instructions: Prob. Points Score possible 1 25 2 25 3 25 4 25 TOTAL 75 (BEST 3) Read eac problem carefully.
More informationQuantum Numbers and Rules
OpenStax-CNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.
More informationCalculus I Homework: The Derivative as a Function Page 1
Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap.
More informationUniversity Mathematics 2
University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 6. Differential Calculus 6.. Differentiation from First Principles. In tis capter, we will introduce
More informationTest 2 Review. 1. Find the determinant of the matrix below using (a) cofactor expansion and (b) row reduction. A = 3 2 =
Test Review Find te determinant of te matrix below using (a cofactor expansion and (b row reduction Answer: (a det + = (b Observe R R R R R R R R R Ten det B = (((det Hence det Use Cramer s rule to solve:
More informationNotes on wavefunctions II: momentum wavefunctions
Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles
More informationA = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
More informationA Reconsideration of Matter Waves
A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,
More informationRecall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if
Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions
More informationAMS 147 Computational Methods and Applications Lecture 09 Copyright by Hongyun Wang, UCSC. Exact value. Effect of round-off error.
Lecture 09 Copyrigt by Hongyun Wang, UCSC Recap: Te total error in numerical differentiation fl( f ( x + fl( f ( x E T ( = f ( x Numerical result from a computer Exact value = e + f x+ Discretization error
More informationSection 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
More informationLines, Conics, Tangents, Limits and the Derivative
Lines, Conics, Tangents, Limits and te Derivative Te Straigt Line An two points on te (,) plane wen joined form a line segment. If te line segment is etended beond te two points ten it is called a straigt
More informationThe Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
More informationChapter 9 - Solved Problems
Capter 9 - Solved Problems Solved Problem 9.. Consider an internally stable feedback loop wit S o (s) = s(s + ) s + 4s + Determine weter Lemma 9. or Lemma 9. of te book applies to tis system. Solutions
More informationChapter 2. Limits and Continuity 16( ) 16( 9) = = 001. Section 2.1 Rates of Change and Limits (pp ) Quick Review 2.1
Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 969) Quick Review..... f ( ) ( ) ( ) 0 ( ) f ( ) f ( ) sin π sin π 0 f ( ). < < < 6. < c c < < c 7. < < < < < 8. 9. 0. c < d d < c
More informationA: Derivatives of Circular Functions. ( x) The central angle measures one radian. Arc Length of r
4: Derivatives of Circular Functions an Relate Rates Before we begin, remember tat we will (almost) always work in raians. Raians on't ivie te circle into parts; tey measure te size of te central angle
More informationDerivation Of The Schwarzschild Radius Without General Relativity
Derivation Of Te Scwarzscild Radius Witout General Relativity In tis paper I present an alternative metod of deriving te Scwarzscild radius of a black ole. Te metod uses tree of te Planck units formulas:
More information1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.
Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of
More informationHow to Find the Derivative of a Function: Calculus 1
Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te
More informationMTH-112 Quiz 1 Name: # :
MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation
More informationCalculus I Practice Exam 1A
Calculus I Practice Exam A Calculus I Practice Exam A Tis practice exam empasizes conceptual connections and understanding to a greater degree tan te exams tat are usually administered in introductory
More informationSFU UBC UNBC Uvic Calculus Challenge Examination June 5, 2008, 12:00 15:00
SFU UBC UNBC Uvic Calculus Callenge Eamination June 5, 008, :00 5:00 Host: SIMON FRASER UNIVERSITY First Name: Last Name: Scool: Student signature INSTRUCTIONS Sow all your work Full marks are given only
More informationSection 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is
Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit
More informationINTRODUCTION TO CALCULUS LIMITS
Calculus can be divided into two ke areas: INTRODUCTION TO CALCULUS Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and
More informationChapter 2 Limits and Continuity
4 Section. Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 6) Quick Review.. f () ( ) () 4 0. f () 4( ) 4. f () sin sin 0 4. f (). 4 4 4 6. c c c 7. 8. c d d c d d c d c 9. 8 ( )(
More informationSinusoidal Steady State Analysis (AC Analysis) Part I
Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University elc.n102.eng@gmail.com http://scholar.cu.edu.eg/refky/
More information4.2 - Richardson Extrapolation
. - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence
More information1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).
. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use
More informationSingle Phase Parallel AC Circuits
Single Phase Parallel AC Circuits 1 Single Phase Parallel A.C. Circuits (Much of this material has come from Electrical & Electronic Principles & Technology by John Bird) n parallel a.c. circuits similar
More informationName: Sept 21, 2017 Page 1 of 1
MATH 111 07 (Kunkle), Eam 1 100 pts, 75 minutes No notes, books, electronic devices, or outside materials of an kind. Read eac problem carefull and simplif our answers. Name: Sept 21, 2017 Page 1 of 1
More informationChapter 33. Alternating Current Circuits
Chapter 33 Alternating Current Circuits 1 Capacitor Resistor + Q = C V = I R R I + + Inductance d I Vab = L dt AC power source The AC power source provides an alternative voltage, Notation - Lower case
More informationWeek #15 - Word Problems & Differential Equations Section 8.2
Week #1 - Word Problems & Differential Equations Section 8. From Calculus, Single Variable by Huges-Hallett, Gleason, McCallum et. al. Copyrigt 00 by Jon Wiley & Sons, Inc. Tis material is used by permission
More information1. (a) 3. (a) 4 3 (b) (a) t = 5: 9. (a) = 11. (a) The equation of the line through P = (2, 3) and Q = (8, 11) is y 3 = 8 6
A Answers Important Note about Precision of Answers: In many of te problems in tis book you are required to read information from a grap and to calculate wit tat information. You sould take reasonable
More informationMaths for Computer Graphics
Trigonometry Is concerned wit te analysis of triangles. Degrees and radians Te degree (or sexagesimal unit of measure derives from defining one complete rotation as 360. Eac degree divides into 60 minutes,
More informationLast lecture (#4): J vortex. J tr
Last lecture (#4): We completed te discussion of te B-T pase diagram of type- and type- superconductors. n contrast to type-, te type- state as finite resistance unless vortices are pinned by defects.
More informationSome Review Problems for First Midterm Mathematics 1300, Calculus 1
Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,
More information2.8 The Derivative as a Function
.8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open
More information3 Minority carrier profiles (the hyperbolic functions) Consider a
Microelectronic Devices and Circuits October 9, 013 - Homework #3 Due Nov 9, 013 1 Te pn junction Consider an abrupt Si pn + junction tat as 10 15 acceptors cm -3 on te p-side and 10 19 donors on te n-side.
More informationIEOR 165 Lecture 10 Distribution Estimation
IEOR 165 Lecture 10 Distribution Estimation 1 Motivating Problem Consider a situation were we ave iid data x i from some unknown distribution. One problem of interest is estimating te distribution tat
More information