ECE 451 Automated Microwave Measurements. TRL Calibration

Size: px
Start display at page:

Download "ECE 451 Automated Microwave Measurements. TRL Calibration"

Transcription

1 utomted Microwve Mesurements Clibrtion Jose E. Schutt-ine Electricl & Computer Engineering University of Illinois Copyright by Jose E. Schutt ine, ll ights eserved

2 Coxil Microstrip rnsition ord with trces Center pin Flnge-mount connector screw -shped support Copyright by Jose E. Schutt ine, ll ights eserved

3 Coxil Microstrip rnsition In SM C C C SM C Out Equivlent Circuit D Plot Copyright by Jose E. Schutt ine, ll ights eserved 3

4 With prsitics No prsitics Copyright by Jose E. Schutt ine, ll ights eserved 4

5 CIION SCHEME coxil connector coxil connector DU microstrip microstrip Wnt to mesure DU only nd need to remove the effect of cox-to-microstrip trnsitions. Use clibrtion Copyright by Jose E. Schutt ine, ll ights eserved 5

6 Error ox Modeling model for the different error boxes cn be implemented Mesurement W Plnes W Port Error ox Γ Γ Error ox Port Error boxes nd ccount for the trnsition prsitics nd the electricl lengths of the microstrip. Mke three stndrds: hru, ine nd eflect Copyright by Jose E. Schutt ine, ll ights eserved 6

7 Step - HU Clibrtion connect thru t b = t b Copyright by Jose E. Schutt ine, ll ights eserved 7

8 Step - INE Clibrtion connect line (Note: difference in length between thru nd line) INE d INE b Copyright by Jose E. Schutt ine, ll ights eserved 8

9 Step 3 - EFEC Clibrtion connect reflect EFEC b Copyright by Jose E. Schutt ine, ll ights eserved 9

10 Mesurement Comprison Mesured S of Microstrip Unknown eltive to OUCHSONE Models 5 PO EX. dt compred to =.808 nh model dt compred to =.948 nh model eltive Mgnitude, d 0-5 PO EX. clibrtion clibrtion Frequency, GHz Copyright by Jose E. Schutt ine, ll ights eserved 0

11 Mesurement Comprison 0 Mesured Dt for Microstrip Unknown Mesured 0/8/94-5 S (d) with clibrtion with 7 ps port ext. (inc. brrel) Frequency, GHz Copyright by Jose E. Schutt ine, ll ights eserved

12 Derivtion Objectives - Obtin network prmeters of error boxes nd - emove their effects in subsequent mesurements Copyright by Jose E. Schutt ine, ll ights eserved

13 Model for eflect S S b S S Γ Γ S S b S S b = 0 b = 0 Mesurements Copyright by Jose E. Schutt ine, ll ights eserved 3

14 Model for hru S S b S S S S b S S b = 0 b = 0 b = 0 b = 0 4 Mesurements Copyright by Jose E. Schutt ine, ll ights eserved 4

15 Model for ine S e -γ S b S S S S b S e -γ S b = 0 b = 0 b = 0 b = 0 4 Mesurements Copyright by Jose E. Schutt ine, ll ights eserved 5

16 Use (or ) Prmeters Using prmeters (trnsfer prmeters), we cn show tht if b = S+ S b = S + S b Δ S b = S S Δ= S S S S Δ = S S S Copyright by Jose E. Schutt ine, ll ights eserved 6

17 Derivtion he mesurement mtrix M is just the product of the mtrices of the error boxes nd the unknown DU or = M = M et be written s r r b = r r r = c is similrly written s ρ ρ α β = ρ ρ ρ = γ he inverse of is b = r bc c Copyright by Jose E. Schutt ine, ll ights eserved 7

18 nd the inverse of is Derivtion β = ρ α βγ γ α he mtrix of the DU is then found from b β = M r c ρ α β b γ c γ α α Note tht lthough there re eight terms in the error boxes, only seven quntities re needed to find. hey re, b, c, α, β, γ, nd r ρ From the mesurement of the through nd of the line, seven quntities will be found. hey re b, c/, β/ α, γ, r ρ, α nd e γl In ddition to the seven quntities, if were found, the solution would be complete. et us first find the bove seven quntities. he idel through hs n mtrix which is the x unit mtrix. he mesured mtrix with the through connected will be denoted by nd is given by = Where nd re the mtrices of the error box nd respectively. With the line connected, the mesured mtrix will be denoted by D nd is equl to Copyright by Jose E. Schutt ine, ll ights eserved 8

19 = D where is the mtrix of the line Derivtion Now = so tht = D D = Define = D Which when substituted into the bove equtions results in = he mtrix is known from mesurements nd will be written s t t = t t γ l e 0 = + γ l 0 e, since the line is non-reflecting Copyright by Jose E. Schutt ine, ll ights eserved 9

20 Derivtion is unknown nd ws written s r r b = r r r = c similrly ws written s ρ ρ α β = ρ ρ ρ = γ eclling = nd writing the mtrices results in γ l t t b b e 0 t t c = c 0 e + γ l Next, writing out the four equtions gives: Copyright by Jose E. Schutt ine, ll ights eserved 0

21 Derivtion l t + t c= e γ l t + t c= ce γ l t b+ t = be + γ t b+ t = e +γ l Dividing the first of the bove eqution by the second results in t + t t+ tc = = c t+ t c c t + t c which gives qudrtic eqution for /c t + ( t t ) t = 0 c c Dividing the third eqution in the group by the fourth results in Copyright by Jose E. Schutt ine, ll ights eserved

22 Derivtion t b+ t t b+ t = b which gives the nlogous qudrtic eqution for b s ( ) t b + t t b t = 0 Dividing the fourth eqution in the group by the second results in γ t b+ t t b+ t e = c = t + tc t + t c Since e γ is not equl to, b nd c/ re distinct roots of the qudrtic eqution. he following discussion will enble the choice of the root. Now b=r /r =S nd r S S = = S c r S Copyright by Jose E. Schutt ine, ll ights eserved

23 Derivtion For well designed trnsition between cox nd the non-cox S, S << which yields b << nd /c >>. herefore, b c which determines the choice of the root eclling = ( det )( det ) = ( det )( det ) or ( ) ( ) det = det = so tht tt tt = which implies tht there re only three independent ij. hen there re only three independent results, e.g. b, /c, nd e γ. Copyright by Jose E. Schutt ine, ll ights eserved 3

24 Derivtion Now let us find four more quntities Now b α β d e rρ g c = = = γ f b b c = bc c So tht or r ρ r α β g b d e γ = bc c f ρ α β g d bf e b γ = bc f cd ce Copyright by Jose E. Schutt ine, ll ights eserved 4

25 from which we cn extrct We lso hve Derivtion c e ce rρ = g = g bc c b α β d bf e b γ = ce f cd ce from which we obtin nd γ = c f d c e β e b = α d bf Copyright by Jose E. Schutt ine, ll ights eserved 5

26 nd α = d bf c e Derivtion he dditionl four quntities found re β/α, γ, r ρ nd α. o complete the solution, one needs to find. et the reflection mesurement through error box be w. hen w = Γ + b cγ + which my be solved for in terms of the known b nd /c s = w b c Γ w We need method to determine. Use the mesurement for the reflect from through the error box. et w denote the mesurement w SSΓ S ΔΓ = S + = S Γ S Γ Copyright by Jose E. Schutt ine, ll ights eserved 6

27 Derivtion or w = w ρ ρ Γ ρ ρ + ρ ρ Γ αγ γ = βγ α my be found in terms of γ nd β/α s α = w + γ β Γ + w α ecll w b = c Γ w Copyright by Jose E. Schutt ine, ll ights eserved 7

28 so tht = α w w Derivtion β + w b α + γ c w From erlier so tht or α d bf = c e w b d bf = w β + w α + γ c c w e =± w w b d bf β + w α c c + γ w e which determines to within ± sign. Copyright by Jose E. Schutt ine, ll ights eserved 8

29 Derivtion Γ w b = c w So if Γ is known to within ± then my be determined s well. Clibrtion is complete nd we cn now proceed to the mesurement of the DU. From erlier, the mtrix of the DU is found from b β = M r c ρ α β b γ c γ α α in which ll the terms hve now been determined. Copyright by Jose E. Schutt ine, ll ights eserved 9

ECE 451 Automated Microwave Measurements. TRL Calibration

ECE 451 Automated Microwave Measurements. TRL Calibration ECE 45 utomted Microwve Mesurements L Clibrtion Jose E. Schutt-ine Electricl & Computer Engineering University of Illinois jschutt@emlb.uiuc.edu ECE 45 Jose Schutt ine Coxil Microstrip rnsition ord with

More information

Chapter Direct Method of Interpolation More Examples Electrical Engineering

Chapter Direct Method of Interpolation More Examples Electrical Engineering Chpter. Direct Method of Interpoltion More Emples Electricl Engineering Emple hermistors re used to mesure the temperture of bodies. hermistors re bsed on mterils chnge in resistnce with temperture. o

More information

A Matrix Algebra Primer

A Matrix Algebra Primer A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,

More information

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

More information

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in

More information

Best Approximation in the 2-norm

Best Approximation in the 2-norm Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

More information

ODE: Existence and Uniqueness of a Solution

ODE: Existence and Uniqueness of a Solution Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

HW3, Math 307. CSUF. Spring 2007.

HW3, Math 307. CSUF. Spring 2007. HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem

More information

The Algebra (al-jabr) of Matrices

The Algebra (al-jabr) of Matrices Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense

More information

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the

More information

Chapter Direct Method of Interpolation More Examples Civil Engineering

Chapter Direct Method of Interpolation More Examples Civil Engineering Chpter 5. Direct Method of Interpoltion More Exmples Civil Engineering Exmple o mximie ctch of bss in lke, it is suggested to throw the line to the depth of the thermocline. he chrcteristic feture of this

More information

Numerical Linear Algebra Assignment 008

Numerical Linear Algebra Assignment 008 Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com

More information

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)

P 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0) 1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this

More information

AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h

AM1 Mathematical Analysis 1 Oct Feb Exercises Lecture 3. sin(x + h) sin x h cos(x + h) cos x h AM Mthemticl Anlysis Oct. Feb. Dte: October Exercises Lecture Exercise.. If h, prove the following identities hold for ll x: sin(x + h) sin x h cos(x + h) cos x h = sin γ γ = sin γ γ cos(x + γ) (.) sin(x

More information

Elements of Matrix Algebra

Elements of Matrix Algebra Elements of Mtrix Algebr Klus Neusser Kurt Schmidheiny September 30, 2015 Contents 1 Definitions 2 2 Mtrix opertions 3 3 Rnk of Mtrix 5 4 Specil Functions of Qudrtic Mtrices 6 4.1 Trce of Mtrix.........................

More information

NOTE ON TRACES OF MATRIX PRODUCTS INVOLVING INVERSES OF POSITIVE DEFINITE ONES

NOTE ON TRACES OF MATRIX PRODUCTS INVOLVING INVERSES OF POSITIVE DEFINITE ONES Journl of pplied themtics nd Computtionl echnics 208, 7(), 29-36.mcm.pcz.pl p-issn 2299-9965 DOI: 0.752/jmcm.208..03 e-issn 2353-0588 NOE ON RCES OF RIX PRODUCS INVOLVING INVERSES OF POSIIVE DEFINIE ONES

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

df dt f () b f () a dt

df dt f () b f () a dt Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem

More information

Math 33A Discussion Example Austin Christian October 23, Example 1. Consider tiling the plane by equilateral triangles, as below.

Math 33A Discussion Example Austin Christian October 23, Example 1. Consider tiling the plane by equilateral triangles, as below. Mth 33A Discussion Exmple Austin Christin October 3 6 Exmple Consider tiling the plne by equilterl tringles s below Let v nd w be the ornge nd green vectors in this figure respectively nd let {v w} be

More information

fractions Let s Learn to

fractions Let s Learn to 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

More information

#6A&B Magnetic Field Mapping

#6A&B Magnetic Field Mapping #6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

More information

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

Matrices and Determinants

Matrices and Determinants Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

More information

Network Analysis and Synthesis. Chapter 5 Two port networks

Network Analysis and Synthesis. Chapter 5 Two port networks Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on

More information

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.

Space Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space. Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)

More information

Quadratic Forms. Quadratic Forms

Quadratic Forms. Quadratic Forms Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information

Lecture 14: Quadrature

Lecture 14: Quadrature Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Department of Electrical and Computer Engineering, Cornell University. ECE 4070: Physics of Semiconductors and Nanostructures.

Department of Electrical and Computer Engineering, Cornell University. ECE 4070: Physics of Semiconductors and Nanostructures. Deprtment of Electricl nd Computer Engineering, Cornell University ECE 4070: Physics of Semiconductors nd Nnostructures Spring 2014 Exm 2 ` April 17, 2014 INSTRUCTIONS: Every problem must be done in the

More information

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es

More information

1 Linear Least Squares

1 Linear Least Squares Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving

More information

MATRICES AND VECTORS SPACE

MATRICES AND VECTORS SPACE MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN -SPACE AND -SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR

More information

Results on Planar Near Rings

Results on Planar Near Rings Interntionl Mthemticl Forum, Vol. 9, 2014, no. 23, 1139-1147 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/imf.2014.4593 Results on Plnr Ner Rings Edurd Domi Deprtment of Mthemtics, University

More information

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system. Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous

More information

Chapter Direct Method of Interpolation More Examples Chemical Engineering

Chapter Direct Method of Interpolation More Examples Chemical Engineering hter 5. Direct Method of Interoltion More Exmles hemicl Engineering Exmle To find how much het is required to bring kettle of wter to its boiling oint, you re sked to clculte the secific het of wter t

More information

MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK. 10th Grade Unit 5 MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

Best Approximation. Chapter The General Case

Best Approximation. Chapter The General Case Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given

More information

DIRECT CURRENT CIRCUITS

DIRECT CURRENT CIRCUITS DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.

The Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve. Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F

More information

Introduction To Matrices MCV 4UI Assignment #1

Introduction To Matrices MCV 4UI Assignment #1 Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be

More information

Using air lines as references for VNA phase measurements

Using air lines as references for VNA phase measurements Using ir lines s references for VNA phse mesurements Stephen Protheroe nd Nick Ridler Electromgnetics Tem, Ntionl Physicl Lbortory, UK Emil: Stephen.protheroe@npl.co.uk Abstrct Air lines re often used

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Notes on length and conformal metrics

Notes on length and conformal metrics Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued

More information

4 The dynamical FRW universe

4 The dynamical FRW universe 4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which

More information

Student Activity 3: Single Factor ANOVA

Student Activity 3: Single Factor ANOVA MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether

More information

Abstract inner product spaces

Abstract inner product spaces WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

Chapter 18 Two-Port Circuits

Chapter 18 Two-Port Circuits Cpter 8 Two-Port Circuits 8. Te Terminl Equtions 8. Te Two-Port Prmeters 8.3 Anlysis of te Terminted Two-Port Circuit 8.4 nterconnected Two-Port Circuits Motivtion Tévenin nd Norton equivlent circuits

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

AP Calculus Multiple Choice: BC Edition Solutions

AP Calculus Multiple Choice: BC Edition Solutions AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this

More information

Joint distribution. Joint distribution. Marginal distributions. Joint distribution

Joint distribution. Joint distribution. Marginal distributions. Joint distribution Joint distribution To specify the joint distribution of n rndom vribles X 1,...,X n tht tke vlues in the smple spces E 1,...,E n we need probbility mesure, P, on E 1... E n = {(x 1,...,x n ) x i E i, i

More information

Lesson Notes: Week 40-Vectors

Lesson Notes: Week 40-Vectors Lesson Notes: Week 40-Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples

More information

Math 231E, Lecture 33. Parametric Calculus

Math 231E, Lecture 33. Parametric Calculus Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider

More information

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements: Mtrices Elementry Mtrix Theory It is often desirble to use mtrix nottion to simplify complex mthemticl expressions. The simplifying mtrix nottion usully mkes the equtions much esier to hndle nd mnipulte.

More information

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

More information

Section 3.2 Maximum Principle and Uniqueness

Section 3.2 Maximum Principle and Uniqueness Section 3. Mximum Principle nd Uniqueness Let u (x; y) e smooth solution in. Then the mximum vlue exists nd is nite. (x ; y ) ; i.e., M mx fu (x; y) j (x; y) in g Furthermore, this vlue cn e otined y point

More information

Chapter 14. Matrix Representations of Linear Transformations

Chapter 14. Matrix Representations of Linear Transformations Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

Determinants Chapter 3

Determinants Chapter 3 Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!

More information

POLYPHASE CIRCUITS. Introduction:

POLYPHASE CIRCUITS. Introduction: POLYPHASE CIRCUITS Introduction: Three-phse systems re commonly used in genertion, trnsmission nd distribution of electric power. Power in three-phse system is constnt rther thn pulsting nd three-phse

More information

TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS

TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

5.7 Improper Integrals

5.7 Improper Integrals 458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the

More information

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions

AQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic

More information

APPROXIMATE INTEGRATION

APPROXIMATE INTEGRATION APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be

More information

Chapter 3 Polynomials

Chapter 3 Polynomials Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling

More information

Chapter 2. Determinants

Chapter 2. Determinants Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

More information

ntegration (p3) Integration by Inspection When differentiating using function of a function or the chain rule: If y = f(u), where in turn u = f(x)

ntegration (p3) Integration by Inspection When differentiating using function of a function or the chain rule: If y = f(u), where in turn u = f(x) ntegrtion (p) Integrtion by Inspection When differentiting using function of function or the chin rule: If y f(u), where in turn u f( y y So, to differentite u where u +, we write ( + ) nd get ( + ) (.

More information

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio. Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number

More information

Chapter 8: Methods of Integration

Chapter 8: Methods of Integration Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

ad = cb (1) cf = ed (2) adf = cbf (3) cf b = edb (4)

ad = cb (1) cf = ed (2) adf = cbf (3) cf b = edb (4) 10 Most proofs re left s reding exercises. Definition 10.1. Z = Z {0}. Definition 10.2. Let be the binry reltion defined on Z Z by, b c, d iff d = cb. Theorem 10.3. is n equivlence reltion on Z Z. Proof.

More information

5.4 The Quarter-Wave Transformer

5.4 The Quarter-Wave Transformer 3/4/7 _4 The Qurter Wve Trnsformer /.4 The Qurter-Wve Trnsformer Redg Assignment: pp. 73-76, 4-43 By now you ve noticed tht qurter-wve length of trnsmission le ( = λ 4, β = π ) ppers often microwve engeerg

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

US01CMTH02 UNIT Curvature

US01CMTH02 UNIT Curvature Stu mteril of BSc(Semester - I) US1CMTH (Rdius of Curvture nd Rectifiction) Prepred by Nilesh Y Ptel Hed,Mthemtics Deprtment,VPnd RPTPScience College US1CMTH UNIT- 1 Curvture Let f : I R be sufficiently

More information

Graduate Students do all problems. Undergraduate students choose three problems.

Graduate Students do all problems. Undergraduate students choose three problems. OPTI 45/55 Midterm Due: Februr, Grdute Students do ll problems. Undergrdute students choose three problems.. Google Erth is improving the resolution of its globl mps with dt from the SPOT5 stellite. The

More information

CHAPTER 4a. ROOTS OF EQUATIONS

CHAPTER 4a. ROOTS OF EQUATIONS CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment

More information

Discrete Least-squares Approximations

Discrete Least-squares Approximations Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

More information

Coalgebra, Lecture 15: Equations for Deterministic Automata

Coalgebra, Lecture 15: Equations for Deterministic Automata Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined

More information

Chapter 1: Fundamentals

Chapter 1: Fundamentals Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,

More information

1 1D heat and wave equations on a finite interval

1 1D heat and wave equations on a finite interval 1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion

More information

Introduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices

Introduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices Introduction to Determinnts Remrks The determinnt pplies in the cse of squre mtrices squre mtrix is nonsingulr if nd only if its determinnt not zero, hence the term determinnt Nonsingulr mtrices re sometimes

More information

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

More information

Solutions to Assignment #8

Solutions to Assignment #8 Mth 1 Numericl Anlysis (Bueler) December 9, 29 Solutions to Assignment #8 Problems 64, exercise 14: The nswer turns out to be yes, which mens tht I hve to be orgnized in writing it up There re lot of fcts

More information

Lecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c

Lecture Outline. Dispersion Relation Electromagnetic Wave Polarization 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3c Course Instructor Dr. Rymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mil: rcrumpf@utep.edu EE 4347 Applied Electromgnetics Topic 3c Wve Dispersion & Polriztion Wve Dispersion These notes & Polriztion

More information

Chapter 10: Symmetrical Components and Unbalanced Faults, Part II

Chapter 10: Symmetrical Components and Unbalanced Faults, Part II Chpter : Symmetricl Components nd Unblnced Fults, Prt.4 Sequence Networks o Loded Genertor n the igure to the right is genertor supplying threephse lod with neutrl connected through impednce n to ground.

More information

Correct answer: 0 m/s 2. Explanation: 8 N

Correct answer: 0 m/s 2. Explanation: 8 N Version 001 HW#3 - orces rts (00223) 1 his print-out should hve 15 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Angled orce on Block 01 001

More information

Natural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring

Natural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring More generlly, we define ring to be non-empty set R hving two binry opertions (we ll think of these s ddition nd multipliction) which is n Abelin group under + (we ll denote the dditive identity by 0),

More information

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon. EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work

More information

Math 270A: Numerical Linear Algebra

Math 270A: Numerical Linear Algebra Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner

More information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

Matrices, Moments and Quadrature, cont d

Matrices, Moments and Quadrature, cont d Jim Lmbers MAT 285 Summer Session 2015-16 Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl

More information