ECE Connections: What do Roots of Unity have to do with OP-AMPs? Louis Scharf, Colorado State University PART 1: Why Complex?
|
|
- Ariel Nancy Cross
- 5 years ago
- Views:
Transcription
1 ECE Conncion: Wha do Roo of Uni hav o do wih OP-AMP? Loui Scharf, Colorado Sa Univri PART : Wh Compl?. Curioi, M favori curioi i : π π ( ) π π ECE Conncion: Colorado Sa Univri Ocobr 007
2 . Quion, Bu wha abou h quion bfor u, If vrhing w build and maur i ral, wh do w u compl anali? And, wha do h compl roo of uni hav o do wih ral OP-AMP? 3. Anwr. Y, vrhing w build and maur i ral. Bu vrhing w build ha wo ral channl (ihr acual or virual) and h wo channl ar convninl rprnd wih on compl channl. Thi i h boomlin anwr. For man of h ampl o follow, compl anali implifi our lif. For on of h ampl, hr would b no lif wihou compl anali. ECE Conncion: Colorado Sa Univri Ocobr 007
3 4. Th Gnral Ida, Rplac wo ral channl wih on compl channl. : ral z : compl : ral z + θ r z z Th ral par of cod for (or and for) h -channl, and h imaginar par of cod for h -channl. Wha do h doubl arrow man? (, ) z : z + r θ : r +, θ arcan / (, ) z : R z z + z *, z z * ; r coθ, r inθ z + i h unral opraor. Or, i i h unimaginabl opraor? ECE Conncion: Colorado Sa Univri Ocobr 007
4 z : 4. Th Gnral Ida (coninud): : ral : ral compl z + Ral mporal mod (or ignvcor) Ral mporal mod (or ignvcor) coω inω Compl mporal mod (or ignvcor) coω + inω ω : ral : ral z : compl Ral paial mod (or -coordina of a fild Ral paial mod (or -coordina of a fild Compl rprnaion of a dimnional fild : ral : ral z : compl Ral caual par of a ignal Ral ani-caual par of a ignal Compl rprnaion of a wo-idd ignal ECE Conncion: Colorado Sa Univri Ocobr 007
5 5. Eampl : Phaor (ECE 0, 3, 33, 34, 444, 457). Phaor rprn ral ignal. ( On phaor rp. Aco( ω R A θ + θ ) ω ω θ A Acoθ coω A θ ω + A Ainθ inω θ +ω ω ω A θ Two phaor rp. coω A θ u coθ v inθ inω ( Ral modulaor rp. w u + ECE Conncion: Colorado Sa Univri Ocobr 007 ω v co ω + z( inω Compl modulaor rp.
6 5. Eampl : Modulaor and Dmodulaor (ECE 3, 3, 33, 33, 4, 43, 444, 457). Phaor gnraliz o im varing phaor. coω coω u( LPF u( v( ( LPF v( inω TX: Baband informaion-baring ignal, modulad o RF inω RX: RF ignal dmodulad o baband o rcovr info. w( R ( LPF w( ω ECE Conncion: Colorado Sa Univri Ocobr 007 ω ω w ( u( + v( co ω + inω Compl mod/dmod
7 ECE Conncion: Colorado Sa Univri Ocobr Eampl 3: -Dimnional Fild and Polarizaion (ECE 34, 34, 444, 457). Phaor alo gnraliz o mimachd phaor. Th produc Liaou figur, or llip. Sum of wo linarl polarizd fild Sum of lf and righ circularl polarizd fild B B A A A A E E z ω ϕ ω ϕ ω θ ω θ θ ω θ ω Im R ) in ( ) co( ) ( ) ( ) ( z( Imag ( -coordina R ( -coordina
8 7. Eampl 4: Filr Dign (ECE 0, 3, 3, 33, 33). Th n pol will hav o b organizd ino caual abl pol and ani-caual abl pol. Thi ugg ha righ-half plan pol ar no all bad. Mor o com. End of Par, dmonraing h powr of compl rprnaion for rprning ral ignal and ral fild. ECE Conncion: Colorado Sa Univri Ocobr 007
9 PART : Polnomial, Roo of Uni, ODE, and OP-AMP +. : Richard Fman (Nobl QED) likd h quaion π + 0 0,,,, π bcau i id oghr. Bu w know zro of h fir ordr pol +. i u h ingl ECE urn zro ino pol b conrucing a raional ranfr funcion: H ( ) h( u( + In our circui and m cour (ECE 0, 3, 3, 33, 33), w dcrib filr b h ranfr quaion. To a H ( ) /( + ) d ( / d + ( ( i o a. π Y ( ) H ( ) X ( ) ( + ) Y ( ) X ( ), which i o a Tha i, b urning zro of a polnomial ino pol of a ranfr funcion, w conruc h cofficin of an ordinar diffrnial quaion from h cofficin of a pol. Thi ugg ha good diffrnial quaion, or good circui, hould b conrucd from good polnomial. Good pol. ECE Conncion: Colorado Sa Univri Ocobr 007
10 + +. : L inch along o h pol, which i h pol ha fir forcd compl numbr ino our licon. Th corrponding ranfr funcion i H ( ) + ( + )( ) / + / + h( in u( Thi i ECE 0, 3, 3, and M 60, 6, 340. W hav variou dcripion of hi ranfr funcion. h( π H ( ω) ω ECE Conncion: Colorado Sa Univri Ocobr 007
11 +. (coninud): From h ranfr quaion, w can driv h ordinar diffrnial quaion for hi m, al an analog compur wiring diagram, and dign an OP-AMP circui: Y ( ) H ( ) X ( ) d ( + ) Y ( ) X ( ) + ( d ( ( d ( / d ( ( 0 Thi i h conncion bwn M 60, 6, 340, ECE 0, 3, 3, 33, 33. ECE Conncion: Colorado Sa Univri Ocobr 007
12 . + (coninud): To h circui dignr, h OP-AMP i h hing. Bu o a m dignr, h ranfr funcion i h hing. X H () Y H X H () H () A a pol, cod for ODE; a a compl impdanc, map ino. ( ( H ( ) X H ( ω) Y H X H ( ω) ( ω ( H ( ω), cald b h ignvalu H ( ω). A a compl frqunc rpon, map h ω ignvcor ino h ignvcor h( h h ( ( A an impul rpon, map ino ( ( h )(. Mak no miak: Th OP-AMP do i. Howvr, h ranfr funcion dcrib h ODE obd b h OP-AMP. Th compl frqunc rpon rval frqunc lcivi b lling u ha i i a if w ar mulipling frqunc componn in h frqunc domain, and h impul rpon plain im domain moohn b lling u ha i i a if w ar convolving ignal in h im domain. ECE Conncion: Colorado Sa Univri Ocobr 007
13 +. (coninud): In circui and m, filring i don in h im domain, and alkd abou in h frqunc domain. W nvr acuall build H ( ω). Rahr, w build a circui o m pcificaion on H ( ω). Bu in cohrn opic, w u ln o Fourir ranform inpu, mulipl b h pcral mak H ( ω), o g Y ( ω) H ( ω) X ( ω), and invr Fourir ranform wih anohr ln o g (. Tha i, in cohrn opic, filring i acuall don in h frqunc domain, and alkd abou in im domain. Th impul rpon i calld h poin prad funcion. h( Thing can g vn fancir in cohrn opic, whr a prim i ud o para frqunc componn ino pac, whr frqunc domain filring i applid o conruc vr hor pul. Thi ECE 457. ECE Conncion: Colorado Sa Univri Ocobr 007
14 . + (coninud): Whr do im domain moohn com from, and frqunc domain lcivi?, ( Anwr: A vr im mu b o mooh ha i, plu i cond drivaiv i zro. Coin do i. For mor gnral diffrnial quaion, linar combinaion of dampd compl ponnial do i. ( ( A om frqunci, h rpon of o an inpu, ω ω ω naml d / d + ω A + A, ha o much druciv inrfrnc bwn h wo componn ha h circui can uppor larg ampliud A whil ming h uni inpu conrain. A h frqunci, h circui or m ha larg gain. Convrl,. So, vn filring i, a boom, an inrfrnc ffc. Inrfrnc i bhind vrhing: focuing in camra, opical imag formaion, polarizd gla, radiaion parn in annna, circui, conrol, and communicaion. 0 A ω ω ω ω A ω ω 0 A 0 ω A ω ECE Conncion: Colorado Sa Univri Ocobr 007
15 . + (coninud): Ar an of h inigh uful for m or channl which do no ob ODE? H () Anwr: Y. W r o modl non-raional ranfr funcion, wih raional approimaion, a illurad blow: G() H G H G H G H G Sri: qualizr for communicaion. (ECE 3, 3, 4, 43) Paralll: modl followr for conrol. (ECE 3, 3, 4, 4) ECE Conncion: Colorado Sa Univri Ocobr 007
16 + n 3. : L g rckl. Thi polnomial ha h nh roo of - for i zro. Th zro could no b idnifid wihou compl anali. n k, k,,..., n H () W lik h LHP pol a pol of a caual and abl filr. Acuall, hr i a lo o lik abou h RHP pol: h ar pol of an ani-caual and abl filr. In fac, bcau of h mmri, if w u h LHP pol o conruc a caual ranfr funcion H (), h RHP pol will * * auomaicall build h ani-caual ranfr funcion H ( ). Thi grouping of pol could no b don wihou compl anali. H () i h ranfr funcion for a caual D/A or ranmi filr in a cll * * phon and a caual D/A or nhi filr in a digial conrollr. H ( ) i h ranfr funcion for an ani-caual A/D or rciv filr in a cll phon and an ani-caual A/D or anali filr in a digial conrollr. Evr cll phon and vr digial conrollr wih i could hav a non-caual A/D. ECE Conncion: Colorado Sa Univri Ocobr
17 + n 3. (coninud) : L conruc h ranfr funcion H () from h LHP pol, and u h compl mmr of h pol o facor H () ino wo cond ordr cion: H ( ) H ( ) H ( ) ζ + ζ Thi facoring could no b don wihou compl anali. Each of h ranfr funcion cod for an ODE: + ( ζ + ) Y ( ) X ( ) d ( / d ζ d( / d + ( ( Each ODE cod for an analog compur wiring diagram: ζ ζ Thi analog compur wiring diagram hould cod for an OP-AMP circui. Thi i ECE 3, 3, 33, 33. ECE Conncion: Colorado Sa Univri Ocobr 007
18 + n 3. (coninud) : L rdraw h analog compur wiring diagram: k k k k Thi i fdback. ECE 3, 3, 33, 33, 4, 4. L build i up, from crach. ECE Conncion: Colorado Sa Univri Ocobr 007
19 + n 3. (coninud) : L idnif an OP-AMP circui wih ach of h uccivl mor complicad fdback diagram. W ar don. ECE Conncion: Colorado Sa Univri Ocobr 007
20 4. Summar : Singl compl channl cod for dual ral channl, hrb clarifing our undranding of modulaion and polarizaion. Roo of uni cod for a ranfr funcion. Thi ranfr funcion ma b facord ino i caual and ani-caual par. Th caual par cod for an ODE. Thi ODE cod for an analog compur wiring diagram. Thi analog compur wiring diagram cod for an OP-AMP circui. Good OP-AMP circui com from good polnomial. You nam h ffc ou ar afr and I will argu for inrfrnc ffc a h undrling bai for wha ou can achiv. Inrfrnc ffc ar rvald b phaor. In raional m of RLC and OP-AMP, h inrfrnc ffc ar drmind b polnomial. ECE Conncion: Colorado Sa Univri Ocobr 007
LaPlace Transform in Circuit Analysis
LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
More informationREPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.
Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[
More informationTransfer function and the Laplace transformation
Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and
More informationInstructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems
Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi
More informationChapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More informationwhere: u: input y: output x: state vector A, B, C, D are const matrices
Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &
More informationCSE 245: Computer Aided Circuit Simulation and Verification
CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy
More informationElementary Differential Equations and Boundary Value Problems
Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationBoyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors
Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar
More informationWave Equation (2 Week)
Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris
More informationPhysics 160 Lecture 3. R. Johnson April 6, 2015
Physics 6 Lcur 3 R. Johnson April 6, 5 RC Circui (Low-Pass Filr This is h sam RC circui w lookd a arlir h im doma, bu hr w ar rsd h frquncy rspons. So w pu a s wav sad of a sp funcion. whr R C RC Complx
More informationSOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz
STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random
More informationWhy Laplace transforms?
MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v
More informationCircuit Transients time
Circui Tranin A Solp 3/29/0, 9/29/04. Inroducion Tranin: A ranin i a raniion from on a o anohr. If h volag and currn in a circui do no chang wih im, w call ha a "ady a". In fac, a long a h volag and currn
More informationLaplace Transforms recap for ccts
Lalac Tranform rca for cc Wha h big ida?. Loo a iniial condiion ron of cc du o caacior volag and inducor currn a im Mh or nodal analyi wih -domain imdanc rianc or admianc conducanc Soluion of ODE drivn
More informationVoltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!
Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr
More informationBoyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues
Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm
More informationH is equal to the surface current J S
Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion
More information2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35
MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h
More informationXV Exponential and Logarithmic Functions
MATHEMATICS 0-0-RE Dirnial Calculus Marin Huard Winr 08 XV Eponnial and Logarihmic Funcions. Skch h graph o h givn uncions and sa h domain and rang. d) ) ) log. Whn Sarah was born, hr parns placd $000
More informationUNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o
More informationSpring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an
More informationChapter 9 The Laplace Transform
Chapr 9 Th Laplac Tranform 熊红凯特聘教授 hp://min.ju.du.cn 电子工程系上海交通大学 7 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationLecture 26: Leapers and Creepers
Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn.
More informationINTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS
adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin
More informationAR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More informationC From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction.
Inducors and Inducanc C For inducors, v() is proporional o h ra of chang of i(). Inducanc (con d) C Th proporionaliy consan is h inducanc, L, wih unis of Hnris. 1 Hnry = 1 Wb / A or 1 V sc / A. C L dpnds
More informationMidterm exam 2, April 7, 2009 (solutions)
Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions
More informationPoisson process Markov process
E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc
More information18.03SC Unit 3 Practice Exam and Solutions
Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care
More information2. Transfer function. Kanazawa University Microelectronics Research Lab. Akio Kitagawa
. ransfr funion Kanazawa Univrsiy Mirolronis Rsarh Lab. Akio Kiagawa . Wavforms in mix-signal iruis Configuraion of mix-signal sysm x Digial o Analog Analog o Digial Anialiasing Digial moohing Filr Prossor
More informationfiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are
MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor
More informationPFC Predictive Functional Control
PFC Prdiciv Funcional Conrol Prof. Car d Prada D. of Sm Enginring and Auomaic Conrol Univri of Valladolid, Sain rada@auom.uva. Oulin A iml a oibl Moivaion PFC main ida An inroducor xaml Moivaion Prdiciv
More information1 Finite Automata and Regular Expressions
1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o
More informationS.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
More informationCHAPTER. Linear Systems of Differential Equations. 6.1 Theory of Linear DE Systems. ! Nullcline Sketching. Equilibrium (unstable) at (0, 0)
CHATER 6 inar Sysms of Diffrnial Equaions 6 Thory of inar DE Sysms! ullclin Skching = y = y y υ -nullclin Equilibrium (unsabl) a (, ) h nullclin y = υ nullclin = h-nullclin (S figur) = + y y = y Equilibrium
More informationFourier Series and Parseval s Relation Çağatay Candan Dec. 22, 2013
Fourir Sris nd Prsvl s Rlion Çğy Cndn Dc., 3 W sudy h m problm EE 3 M, Fll3- in som dil o illusr som conncions bwn Fourir sris, Prsvl s rlion nd RMS vlus. Q. ps h signl sin is h inpu o hlf-wv rcifir circui
More informationGeneral Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract
Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara Email: rajndraprasadrgmi@yahoo.com
More information14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions
4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway
More informationMEM 355 Performance Enhancement of Dynamical Systems A First Control Problem - Cruise Control
MEM 355 Prformanc Enhancmn of Dynamical Sysms A Firs Conrol Problm - Cruis Conrol Harry G. Kwany Darmn of Mchanical Enginring & Mchanics Drxl Univrsiy Cruis Conrol ( ) mv = F mg sinθ cv v +.2v= u 9.8θ
More information6.302 Feedback Systems Recitation : Phase-locked Loops Prof. Joel L. Dawson
6.32 Feedback Syem Phae-locked loop are a foundaional building block for analog circui deign, paricularly for communicaion circui. They provide a good example yem for hi cla becaue hey are an excellen
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More informationPhys463.nb Conductivity. Another equivalent definition of the Fermi velocity is
39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h
More informationEE 350 Signals and Systems Spring 2005 Sample Exam #2 - Solutions
EE 35 Signals an Sysms Spring 5 Sampl Exam # - Soluions. For h following signal x( cos( sin(3 - cos(5 - T, /T x( j j 3 j 3 j j 5 j 5 j a -, a a -, a a - ½, a 3 /j-j -j/, a -3 -/jj j/, a 5 -½, a -5 -½,
More informationChapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu
Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im
More informationMa/CS 6a Class 15: Flows and Bipartite Graphs
//206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d
More informationDISCRETE TIME FOURIER TRANSFORM (DTFT)
DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital
More informationOn the Speed of Heat Wave. Mihály Makai
On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.
More informationCircuits and Systems I
Circuis and Sysms I LECTURE #3 Th Spcrum, Priodic Signals, and h Tim-Varying Spcrum lions@pfl Prof. Dr. Volan Cvhr LIONS/Laboraory for Informaion and Infrnc Sysms Licns Info for SPFirs Slids This wor rlasd
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 30 Signal & Syem Prof. ark Fowler oe Se #34 C-T Tranfer Funcion and Frequency Repone /4 Finding he Tranfer Funcion from Differenial Eq. Recall: we found a DT yem Tranfer Funcion Hz y aking he ZT of
More information( ) C R. υ in RC 1. cos. ,sin. ω ω υ + +
Oscillaors. Thory of Oscillaions. Th lad circui, h lag circui and h lad-lag circui. Th Win Bridg oscillaor. Ohr usful oscillaors. Th 555 Timr. Basic Dscripion. Th S flip flop. Monosabl opraion of h 555
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More informationReview Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )
Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division
More informationDouble Slits in Space and Time
Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an
More informationLecture 4: Laplace Transforms
Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions
More informationChap.3 Laplace Transform
Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl
More informationPartial Fraction Expansion
Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom.
More informationLet s look again at the first order linear differential equation we are attempting to solve, in its standard form:
Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
More information7.4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS *
Andri Tokmakoff, MIT Dparmn of Chmisry, 5/19/5 7-11 7.4 QUANTUM MECANICAL TREATMENT OF FLUCTUATIONS * Inroducion and Prviw Now h origin of frquncy flucuaions is inracions of our molcul (or mor approprialy
More informationMidterm Examination (100 pts)
Econ 509 Spring 2012 S.L. Parn Midrm Examinaion (100 ps) Par I. 30 poins 1. Dfin h Law of Diminishing Rurns (5 ps.) Incrasing on inpu, call i inpu x, holding all ohr inpus fixd, on vnuall runs ino h siuaion
More informationControl System Engineering (EE301T) Assignment: 2
Conrol Sysm Enginring (EE0T) Assignmn: PART-A (Tim Domain Analysis: Transin Rspons Analysis). Oain h rspons of a uniy fdack sysm whos opn-loop ransfr funcion is (s) s ( s 4) for a uni sp inpu and also
More informationwhereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas
Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically
More informationApplied Statistics and Probability for Engineers, 6 th edition October 17, 2016
Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...
More informationFourier. Continuous time. Review. with period T, x t. Inverse Fourier F Transform. x t. Transform. j t
Coninuous im ourir rnsform Rviw. or coninuous-im priodic signl x h ourir sris rprsnion is x x j, j 2 d wih priod, ourir rnsform Wh bou priodic signls? W willl considr n priodic signl s priodic signl wih
More informationECEN620: Network Theory Broadband Circuit Design Fall 2014
ECE60: work Thory Broadbad Circui Dig Fall 04 Lcur 6: PLL Trai Bhavior Sam Palrmo Aalog & Mixd-Sigal Cr Txa A&M Uivriy Aoucm, Agda, & Rfrc HW i du oday by 5PM PLL Trackig Rpo Pha Dcor Modl PLL Hold Rag
More informationVeer Surendra Sai University of Technology, Burla. S u b j e c t : S i g n a l s a n d S y s t e m s - I S u b j e c t c o d e : B E E
Vr Surndra Sai Univriy of Tchnology, Burla Dparmn o f E l c r i c a l & E l c r o n i c E n g g S u b j c : S i g n a l a n d S y m - I S u b j c c o d : B E E - 6 0 5 B r a n c h m r : E E E 5 h m SYLLABUS
More informationPWM-Scheme and Current ripple of Switching Power Amplifiers
axon oor PWM-Sch and Currn rippl of Swiching Powr Aplifir Abrac In hi work currn rippl caud by wiching powr aplifir i analyd for h convnional PWM (pulwidh odulaion) ch and hr-lvl PWM-ch. Siplifid odl for
More informationConsider a system of 2 simultaneous first order linear equations
Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm
More information13.1 Circuit Elements in the s Domain Circuit Analysis in the s Domain The Transfer Function and Natural Response 13.
Chaper 3 The Laplace Tranform in Circui Analyi 3. Circui Elemen in he Domain 3.-3 Circui Analyi in he Domain 3.4-5 The Tranfer Funcion and Naural Repone 3.6 The Tranfer Funcion and he Convoluion Inegral
More informationEXERCISE - 01 CHECK YOUR GRASP
DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc
More informationThis is an example to show you how SMath can calculate the movement of kinematic mechanisms.
Dec :5:6 - Kinemaics model of Simple Arm.sm This file is provided for educaional purposes as guidance for he use of he sofware ool. I is no guaraeed o be free from errors or ommissions. The mehods and
More informationEE Control Systems LECTURE 2
Copyrigh F.L. Lewi 999 All righ reerved EE 434 - Conrol Syem LECTURE REVIEW OF LAPLACE TRANSFORM LAPLACE TRANSFORM The Laplace ranform i very ueful in analyi and deign for yem ha are linear and ime-invarian
More informationInterpolation and Pulse Shaping
EE345S Real-Time Digial Signal Proceing Lab Spring 2006 Inerpolaion and Pule Shaping Prof. Brian L. Evan Dep. of Elecrical and Compuer Engineering The Univeriy of Texa a Auin Lecure 7 Dicree-o-Coninuou
More informationDecline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.
Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing.
More information5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t
AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =
More informationCIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid
More informationMath 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2
Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationChapter 2 The Derivative Business Calculus 99
Chapr Th Drivaiv Businss Calculus 99 Scion 5: Drivaivs of Formulas In his scion, w ll g h rivaiv ruls ha will l us fin formulas for rivaivs whn our funcion coms o us as a formula. This is a vry algbraic
More informationAn Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT
[Typ x] [Typ x] [Typ x] ISSN : 974-7435 Volum 1 Issu 24 BioTchnology 214 An Indian Journal FULL PAPE BTAIJ, 1(24), 214 [15197-1521] A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss LI
More informationECE 3TR4 Communication Systems (Winter 2004)
ECE 3R4 Communicaion Sysms Winr 4 Dr.. Kirubarajan Kiruba ECE Dparmn CRL-5 iruba@mcmasr.ca www.c.mcmasr.ca/~iruba/3r4/3r4.hml Cours Ovrviw Communicaion Sysms Ovrviw Fourir Sris/ransorm Rviw Signals and
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationMaster Thesis Seminar
Mar Thi minar Hlinki Univriy of Thnology Dparmn of Elrial and Commniaion Enginring Commniaion laboraory Mar hi rforman Evalaion of rially Conanad pa-tim Cod by Aboda Abdlla Ali prvior: prof vn-gav Häggman
More informationExponential Sawtooth
ECPE 36 HOMEWORK 3: PROPERTIES OF THE FOURIER TRANSFORM SOLUTION. Exponenial Sawooh: The eaie way o do hi problem i o look a he Fourier ranform of a ingle exponenial funcion, () = exp( )u(). From he able
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationContinous system: differential equations
/6/008 Coious sysm: diffrial quaios Drmiisic modls drivaivs isad of (+)-( r( compar ( + ) R( + r ( (0) ( R ( 0 ) ( Dcid wha hav a ffc o h sysm Drmi whhr h paramrs ar posiiv or gaiv, i.. giv growh or rducio
More information6.8 Laplace Transform: General Formulas
48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy
More informationEE 529 Remote Sensing Techniques. Review
59 Rmo Snsing Tchniqus Rviw Oulin Annna array Annna paramrs RCS Polariaion Signals CFT DFT Array Annna Shor Dipol l λ r, R[ r ω ] r H φ ηk Ilsin 4πr η µ - Prmiiviy ε - Prmabiliy
More informationFrequency Response. Lecture #12 Chapter 10. BME 310 Biomedical Computing - J.Schesser
Frquncy Rspns Lcur # Chapr BME 3 Bimdical Cmpuing - J.Schssr 99 Idal Filrs W wan sudy Hω funcins which prvid frquncy slciviy such as: Lw Pass High Pass Band Pass Hwvr, w will lk a idal filring, ha is,
More informationThe transition:transversion rate ratio vs. the T-ratio.
PhyloMah Lcur 8 by Dan Vandrpool March, 00 opics of Discussion ransiion:ransvrsion ra raio Kappa vs. ransiion:ransvrsion raio raio alculaing h xpcd numbr of subsiuions using marix algbra Why h nral im
More informationResponse of LTI Systems to Complex Exponentials
3 Fourir sris coiuous-im Rspos of LI Sysms o Complx Expoials Ouli Cosidr a LI sysm wih h ui impuls rspos Suppos h ipu sigal is a complx xpoial s x s is a complx umbr, xz zis a complx umbr h or h h w will
More informationNote 6 Frequency Response
No 6 Frqucy Rpo Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada. alyical Exprio
More information( ) ( ) + = ( ) + ( )
Mah 0 Homwork S 6 Soluions 0 oins. ( ps I ll lav i o you vrify ha h omplimnary soluion is : y ( os( sin ( Th guss for h pariular soluion and is drivaivs ar, +. ( os( sin ( ( os( ( sin ( Y ( D 6B os( +
More information