Network Analysis and Synthesis. Chapter 5 Two port networks
|
|
- Zoe Long
- 5 years ago
- Views:
Transcription
1 Network Anlsis nd Snthesis hpter 5 Two port networks
2 . 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 the other hnd hs two pirs of voltge nd current reltionships.
3 The vrile re,,, Two of these re independent vriles, while the rest re dependent vriles. Hence, there re six possile set of equtions tht descrie two port networks. There re vrious ws to write these reltionships Z,, H nd ABD prmeters. We will discuss ech description. 3
4 . Z-prmeters A prticulr set of equtions tht descrie two port networks re Z-prmeter equtions Z Z Z Z n these equtions nd re the dependent vriles, while nd re the independent vriles. 4
5 The individul -prmeters re defined s 0 0 All the -prmeters hve unit of impednce. The individul -prmeters re specified when the current in one of the ports is ero. i.e. open circuit. Hence, sometimes referred s open circuit prmeters
6 Z reltes the voltge nd current in the first port, while Z reltes the voltge nd current in the second port. These re referred s open circuit driving point impednces. Z reltes the voltge on the first port to the current in the second port, while Z reltes the voltge in second port with the current in the first port. These re referred s open circuit trnsfer impednces. 6
7 Exmple Find the open circuit impednces for the following T circuit. Solution: c Z Z Z Z Z Z Z Z Z Z 7
8 Note tht in the previous exmple Z =Z, hence, the circuit is reciprocl. Most pssive time-invrint networks re reciprocl. 8
9 3. -prmeters The -prmeters re defined s The voltges nd t the ports re the independent vriles, where s the currents through the two ports re the dependent vriles. 9
10 The individul -prmeters re specified s All the -prmeters hve unit of dmittnce. The individul -prmeters re specified when the voltge t one of the ports is ero. i.e. short circuit. Hence, sometimes referred s short circuit prmeters. 0 0
11 Exmple Find the -prmeters of the following circuit. Soln.: Using mesh nlsis, we cn write the following equtions s s s s s
12 Solving for nd, we get
13 Exmple 3 Find the -prmeters for the following π circuit. Solution: We short circuit port to find the nd. We short circuit port to find the nd. A B 3
14 4. H prmeters A set of prmeters tht re ver useful in descriing trnsistor circuits re h prmeters. H prmeters re given the following eqution h h h h nd re the independent vriles, wheres nd re dependent vriles. 4
15 The individul h prmeters re descried s h nd h re short circuit prmeters, while h nd h re open circuit prmeters. Note tht h h h =/. 0 0 h h 0 0 h =/ While h nd h re trnsfer prmeters. 5
16 h prmeters re sometimes clled hrid prmeters since the hve oth short circuit prmeters nd open circuit prmeters. 6
17 Exmple 4 For the pi circuit elow, find the h prmeters using open circuit nd short circuit. Solution: Short circuit the second port to find h nd h Open circuit the first port to find h nd h. A A B A A A h h h h
18 5. ABD prmeters or Trnsmission mtrix The ABD prmeters re given in mtrix form s A B D Or in eqution form The mtrix representtion is clled the trnsmission mtrix. The reson we multipl negtive is in most cses the current on the output port is coming out of the port. 8 A B D
19 n these equtions nd re the independent vriles, while nd re the dependent vriles. The individul A, B,, D prmeters re D B A 9
20 Lets descrie the ABD prmeters with - prmeters nd -prmeters. Exercise: prove the ove eqution. 0
21 Exmple 5 Find the ABD prmeters for the pi circuit elow. Solution: A A B B A B D B A
22 6. Reltionship etween two port prmeters Previousl we showed tht some h prmeters cn e given s function of nd prmeters. t turns out tht n prmeter cn e expressed s function of n other prmeter. This is ovious since ll 4 prmeters specif given port network completel.
23 For exmple lets derive the reltionship etween nd prmeters. f we express the nd prmeter equtions using mtrix representtion 3 or, prmeter representtion into the prmeters Sustituting the Hence
24 Tht is, det det The following slide shows the reltionship etween the 4 prmeters. 4
25 5
26 7. nterconnection of two port networks There re three ws two port networks cn e connected scde Prllel Series We will discuss ech 6
27 7. scde connection A cscde connection is when the port of one two port network is connected to port of the second two port network. Note tht = = 7
28 Lets write the ABD (trnsmission) prmeters for the two networks Hence, when two port networks re interconnected in cscde the trnsmission prmeters re multiplied. 8 D B A D B A D B A D B A since nd
29 7. Prllel A prllel connection is when the voltge t port is equl is equl for the two networks nd the voltge t port is equl is equl for the two networks. Note tht = = = + = + 9
30 Lets write the prmeters for the two networks Hence, when two port networks re connected in prllel, the prmeters re dded. 30 nd since nd
31 7.3 Series connection A series connection of two networks is when the current in port of the networks is the sme nd when the current in port of the two networks is the sme. Note tht = = = + = + 3
32 Lets write the prmeters for the two networks Hence, when two port networks re connected in series the prmeters re dded 3 nd since nd
33 8. Trnsfer function using two port prmeters We hve seen how to get trnsfer function using trnsformed network nlsis. Here, we will discuss how to otin trnsfer functions using two port prmeters. There re two rod ctegories: Without lod or source impednce nd With lod nd/or source impednce. 33
34 8. Without lod or source impednce These cn e descried using the prmeters ( or ) lone. For exmple, the open circuit voltge rtio cn e given s We don t del with these most of the time, since our source will hve source impednce nd we mesure our output on lod. 34 or 0
35 8. With lod or source impednce These trnsfer functions re functions of the prmeters (,, h, t) nd the source nd/or lod impednce. To esil find these trnsfer functions we should discuss out circuit representtions of two port prmeters. 35
36 Z prmeter ircuit representtions + Z Z + Z Z - - Z nd Z re controlled voltge sources in series with Z nd Z respectivel.. 36
37 prmeter nd re controlled current sources in prllel with nd respectivel. 37
38 H prmeter + h + h h /h - - h is controlled voltge source while /h is controlled current source. 38
39 The voltge nd current sources relting the vrious dependent vriles with the independent vriles re clled controlled sources. (This is ecuse the re controlled some vrile on the other port). 39
40 Exmple Find the trnsfer mpednce / following network with prmeter. for the Solution: R nserting the second eqution R R 40
41 Exmple Find the trnsfer dmittnce / following network with prmeter. Solution: for the R nserting the second eqution R R R 4
42 Exmple Find the trnsfer function / g for the network terminted t oth ends, shown elow. Solution: g nd R g R R R R R g g R R R 4
43 Exmple Find the input impednce or / following hrid circuit for the 43
44 44
Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices:
3/3/009 ignl Flow Grphs / ignl Flow Grphs Consider comple 3-port microwve network, constructed of 5 simpler microwve devices: 3 4 5 where n is the scttering mtri of ech device, nd is the overll scttering
More informationDesigning Information Devices and Systems I Discussion 8B
Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V
More informationELE B7 Power System Engineering. Unbalanced Fault Analysis
Power System Engineering Unblnced Fult Anlysis Anlysis of Unblnced Systems Except for the blnced three-phse fult, fults result in n unblnced system. The most common types of fults re single lineground
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationChapter 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 informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:
Vectors 1-23-2018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2-dimensionl vectors: (2, 3), ( )
More information1B40 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 informationMatrix 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 informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationIndustrial Electrical Engineering and Automation
CODEN:LUTEDX/(TEIE-719)/1-7/(7) Industril Electricl Engineering nd Automtion Estimtion of the Zero Sequence oltge on the D- side of Dy Trnsformer y Using One oltge Trnsformer on the D-side Frncesco Sull
More information332:221 Principles of Electrical Engineering I Fall Hourly Exam 2 November 6, 2006
2:221 Principles of Electricl Engineering I Fll 2006 Nme of the student nd ID numer: Hourly Exm 2 Novemer 6, 2006 This is closed-ook closed-notes exm. Do ll your work on these sheets. If more spce is required,
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd
More informationResistive Network Analysis
C H A P T E R 3 Resistive Network Anlysis his chpter will illustrte the fundmentl techniques for the nlysis of resistive circuits. The methods introduced re sed on the circuit lws presented in Chpter 2:
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
More informationPOLYPHASE 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 informationDesigning Information Devices and Systems I Spring 2018 Homework 8
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission
More informationDeterminants 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 informationOVER-DETERMINATION IN ACOUSTIC TWO-PORT DATA MEASUREMENT
OVER-DEERMINAION IN ACOUSIC WO-POR DAA MEASUREMEN Sry Allm, Hns Bodén nd Mts Åom he Mrcus Wllenerg Lortory for Sound nd Virtion Reserch Dept. of Aeronuticl nd Vehicle Engineering, KH, SE-0044 Stockholm,
More informationDesigning Information Devices and Systems I Spring 2018 Homework 7
EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should
More information4.1. Probability Density Functions
STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationChapter 4: Techniques of Circuit Analysis. Chapter 4: Techniques of Circuit Analysis
Chpter 4: Techniques of Circuit Anlysis Terminology Node-Voltge Method Introduction Dependent Sources Specil Cses Mesh-Current Method Introduction Dependent Sources Specil Cses Comprison of Methods Source
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More informationCS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation
CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes
More informationDorf, R.C., Wan, Z. T- Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
orf, R.C., Wn,. T- Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000 9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationChapter 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 informationLecture 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 informationFast Boolean Algebra
Fst Boolen Alger ELEC 267 notes with the overurden removed A fst wy to lern enough to get the prel done honorly Printed; 3//5 Slide Modified; Jnury 3, 25 John Knight Digitl Circuits p. Fst Boolen Alger
More informationOn the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationCharacteristic Equation-Based Computation of Thévenin and Norton Equivalent Circuits
IEEJ TANSACTIONS ON EECTICA AND EECTONIC ENGINEEING IEEJ Trns 0; 6: 509 56 Pulished online in Wiley Online irry (wileyonlinelirry.com). DOI:0.00/tee.0689 Chrcteristic Eqution-Bsed Computtion of Thévenin
More informationNote 12. Introduction to Digital Control Systems
Note Introduction to Digitl Control Systems Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd . Introduction A digitl control system is one in which the
More information0.1 THE REAL NUMBER LINE AND ORDER
6000_000.qd //0 :6 AM Pge 0-0- CHAPTER 0 A Preclculus Review 0. THE REAL NUMBER LINE AND ORDER Represent, clssify, nd order rel numers. Use inequlities to represent sets of rel numers. Solve inequlities.
More informationAnalytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
More informationQuadratic 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 informationSection 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 informationset 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 informationAPPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line
APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente
More informationCoalgebra, 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 informationChapter 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 informationChapter 6 Continuous Random Variables and Distributions
Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete
More information5.4 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 informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More informationSection - 2 MORE PROPERTIES
LOCUS Section - MORE PROPERTES n section -, we delt with some sic properties tht definite integrls stisf. This section continues with the development of some more properties tht re not so trivil, nd, when
More informationChapter 3 Single Random Variables and Probability Distributions (Part 2)
Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationz TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability
TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSP-G 6. Trnsform Bsics The definition of the trnsform for digitl signl is: -n X x[ n is complex vrile The trnsform
More informationThis chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationMethods of Analysis and Selected Topics (dc)
Methods of Anlysis nd Selected Topics (dc) 8 Ojectives Become fmilir with the terminl chrcteristics of current source nd how to solve for the voltges nd currents of network using current sources nd/or
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction
Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the
More informationChapter E - Problems
Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current
More information4.5 Signal Flow Graphs
3/9/009 4_5 ignl Flow Grphs.doc / 4.5 ignl Flow Grphs Reding Assignment: pp. 89-97 Q: Using individul device scttering prmeters to nlze comple microwve network results in lot of mess mth! Isn t there n
More informationMath 259 Winter Solutions to Homework #9
Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationSpecial Relativity solved examples using an Electrical Analog Circuit
1-1-15 Specil Reltivity solved exmples using n Electricl Anlog Circuit Mourici Shchter mourici@gmil.com mourici@wll.co.il ISRAE, HOON 54-54855 Introduction In this pper, I develop simple nlog electricl
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationContinuous Random Variables
CPSC 53 Systems Modeling nd Simultion Continuous Rndom Vriles Dr. Anirn Mhnti Deprtment of Computer Science University of Clgry mhnti@cpsc.uclgry.c Definitions A rndom vrile is sid to e continuous if there
More informationDesigning 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 informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationVectors. Introduction. Definition. The Two Components of a Vector. Vectors
Vectors Introduction This pper covers generl description of vectors first (s cn e found in mthemtics ooks) nd will stry into the more prcticl res of grphics nd nimtion. Anyone working in grphics sujects
More informationBoolean algebra.
http://en.wikipedi.org/wiki/elementry_boolen_lger Boolen lger www.tudorgir.com Computer science is not out computers, it is out computtion nd informtion. computtion informtion computer informtion Turing
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationMethods of Analysis and Selected Topics (dc)
8 N A Methods of Anlysis nd Selected Topics (dc) 8. INTRODUCTION The circuits descried in the previous chpters hd only one source or two or more sources in series or prllel present. The step-y-step procedure
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More informationDesigning Information Devices and Systems I Anant Sahai, Ali Niknejad. This homework is due October 19, 2015, at Noon.
EECS 16A Designing Informtion Devices nd Systems I Fll 2015 Annt Shi, Ali Niknejd Homework 7 This homework is due Octoer 19, 2015, t Noon. 1. Circuits with cpcitors nd resistors () Find the voltges cross
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationChapter 2. Random Variables and Probability Distributions
Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationIntroduction to Electronic Circuits. DC Circuit Analysis: Transient Response of RC Circuits
Introduction to Electronic ircuits D ircuit Anlysis: Trnsient esponse of ircuits Up until this point, we hve een looking t the Stedy Stte response of D circuits. StedyStte implies tht nothing hs chnged
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationConvert the NFA into DFA
Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:
More informationM344 - ADVANCED ENGINEERING MATHEMATICS
M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationFundamentals of Electrical Circuits - Chapter 3
Fundmentls of Electricl Circuits Chpter 3 1S. For the circuits shown elow, ) identify the resistors connected in prllel ) Simplify the circuit y replcing prllel connect resistors with equivlent resistor.
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions
More informationPatch Antennas. Chapter Resonant Cavity Analysis
Chpter 4 Ptch Antenns A ptch ntenn is low-profile ntenn consisting of metl lyer over dielectric sustrte nd ground plne. Typiclly, ptch ntenn is fed y microstrip trnsmission line, ut other feed lines such
More informationExploring parametric representation with the TI-84 Plus CE graphing calculator
Exploring prmetric representtion with the TI-84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges -) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More information/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2
SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More information4.6 Numerical Integration
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
More informationBoolean Algebra. Boolean Algebra
Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with: - two inry opertions, commonly denoted y + nd, - unry opertion, usully denoted y ˉ or ~ or, - two elements usully clled zero nd
More informationNetwork Theorems 9.1 INTRODUCTION 9.2 SUPERPOSITION THEOREM
9 Th Network Theorems 9.1 NTRODUCTON This chpter will introduce the importnt fundmentl theorems of network nlysis. ncluded re the superposition, Thévenin s, Norton s, mximum power trnsfer, sustitution,
More information1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.
Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers
More informationResistors. Consider a uniform cylinder of material with mediocre to poor to pathetic conductivity ( )
10/25/2005 Resistors.doc 1/7 Resistors Consider uniform cylinder of mteril with mediocre to poor to r. pthetic conductivity ( ) ˆ This cylinder is centered on the -xis, nd hs length. The surfce re of the
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More information