Impedance Transformation and Parameter Relations

Size: px
Start display at page:

Download "Impedance Transformation and Parameter Relations"

Transcription

1 8/1/18 Cours nstructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 4 mpdanc Transformation and Paramtr Rlations mpdanc Ths Transformation nots may conta & Paramtrs copyrightd matrial obtad undr fair us ruls. Distribution of ths matrials is strictly prohibitd Slid 1 ctur Outl nput mpdanc, Paramtr Rlations Spcial Cass of Tratd Transmission s Shortd l ( = ) Opn circuit l ( = ) Matchd l ( = ) mpdanc Transformation & Paramtrs Slid 1

2 8/1/18 nput mpdanc, mpdanc Transformation & Paramtrs Slid 3 Problm Stup Gnrator Transmission oad g g, Gnrator g nput mpdanc Th put impdanc is th impdanc obsrvd by th gnrator. g Th put impdanc is NOT ncssarily th l s charactristic impdanc or th load impdanc. mpdanc Transformation & Paramtrs Slid 4

3 8/1/18 Animation of mpdanc Transformation m 4 nput impdanc vrts 15 j15 m nput impdanc rpats 1 j1 mpdanc Transformation & Paramtrs Slid 5 Drivation of nput mpdanc, (1 of ) Th rflction coffict at any pot from th load is Backward Wav Forward Wav This mans that from th prspctiv of th gnrator, th rflction gog to th transmission l will chang dpndg on th lngth of th transmission l. This can only happn of th put impdanc to th transmission l is changg. mpdanc Transformation & Paramtrs Slid 6 3

4 8/1/18 Drivation of nput mpdanc, ( of ) W df th impdanc of th l at position to b W prviously wrot () and () as Substitutg our xprssions for () and () givs t maks sns that th impdanc is not a function of voltag a lar systm. mpdanc Transformation & Paramtrs Slid 7 Sanity Chck: nput mpdanc at oad Th put impdanc at th load can b dtrd by sttg = our prvious quation Cancl from th quation. Multiply numrator and dnoator by. Cancl numrator and dnoator. W got th answr w wr xpctg! mpdanc Transformation & Paramtrs Slid 8 4

5 8/1/18 nput mpdanc at Th put impdanc at location is A Not About Sign: Backg away from th load, bcoms ngativ. Howvr, w dfd so stays positiv this quation and for quations that follow. mpdanc Transformation & Paramtrs Slid 9 mpdanc Transformation Formula (1 of ) Rcall that W can liat from th put impdanc quation by substitutg our xprssion for. cosh sh sh cosh cosh sh mpdanc Transformation & Paramtrs Slid 1 5

6 8/1/18 mpdanc Transformation Formula ( of ) Rcall that tanh sh cosh This lts us writ th put impdanc xprssion as sh tanh tanh sh cosh cosh sh cosh sh cosh mpdanc Transformation & Paramtrs Slid 11 nput Transformation for osslss Th losslss l has j Puttg ths valus to our impdanc transformation formula givs tanh j tanh j Rcognig that tanh(j) = jtan(), th xprssion for losslss ls bcoms j tan j tan mpdanc Transformation & Paramtrs Slid 1 6

7 8/1/18 nput mpdanc Rpats for osslss s For losslss ls, th tan function th impdanc transformation quation tlls us that th function is priodic and rpats. Th function rpats vry tgr multipl of. m m,, 3,, 1,,1,, 3,, Rcognig that = /, th abov xprssion lads to m Not: is th wavlngth th transmission l, not th fr spac wavlngth. This mans th put impdanc rpats for vry half wavlngth long th transmission l is. W will rvisit this whn w covr Smith charts, which will giv you a way to visuali th impdanc transformation phnomnon. mpdanc Transformation & Paramtrs Slid 13 Exampl: mpdanc Transformation (1 of 3) A transmission l with 5 charactristic impdanc is connctd to a 1 nf capacitor as th load. f th phas constant of th transmission l is = 6 m -1, what is th put impdanc of a 1 ch sction of l opratg at 4. GH? What quivalnt circuit would th sourc s? Transmission oad 5 1 nf 1 ch mpdanc Transformation & Paramtrs Slid 14 7

8 8/1/18 Exampl: mpdanc Transformation ( of 3) oss was not spcifid so w assum a losslss transmission l. Our impdanc transformation quation is thrfor j tan j tan Th variabls this quation ar cm 1 m 6 m 1 ch ch 1 cm jc j fc j 4.1 s 11 F j.4 mpdanc Transformation & Paramtrs Slid 15 Exampl: mpdanc Transformation (3 of 3) Substitutg ths valus to th impdanc transformation quation givs j.4 j 5 tan jj.4 tan j1.71 Th put impdanc is purly imagary and positiv. Thus, th put impdanc looks lik an ductor to th gnrator. j q q j j f j1.71 j 4.1 s H 4.4 nh mpdanc Transformation & Paramtrs Slid 16 8

9 8/1/18 Paramtr Rlations mpdanc Transformation & Paramtrs Slid 17,, & Trms of SWR and 1 1 SWR SWR1 SWR1 and 1 1 SWR SWR1 SWR1 mpdanc Transformation & Paramtrs Slid 18 9

10 8/1/18 Trms of SWR Th charactristic impdanc can b calculatd from and or and. Th put impdanc rpats as you back away from th load. W can calculat th imum and imum impdanc as SWR SWR mpdanc Transformation & Paramtrs Slid 19 Exampl (1 of 3) A 5 impdanc transmission l is connctd to an antnna with a 7 put impdanc. A sourc provids an put signal of 4 pak to pak. What is th rflction coffict at th antnna? n this cas, th antnna is th load. 7 5 What fraction of th put powr is dlivrd to th antnna? R Dspit th mismatch, almost all powr is still dlivrd to th antnna. This still T 1R % dos not man th antnna will radiat! What is th SWR on th l fdg th antnna? SWR SWR log SWR log db db 1 1 mpdanc Transformation & Paramtrs Slid 1

11 8/1/18 Exampl ( of 3) What is th imum and imum voltag on th l? First, w nd to convrt voltag pak to pak p-p to voltag magnitud. p-p 4 1 Now w ar a position to calculat and. Whn w ar utilig high voltags, w want to b sur will not caus arcg or any othr brakdown problms. What is th imum and imum currnt on th l? A 5 At high powr, w want to b sur will not.833 A caus hatg problms. 5 mpdanc Transformation & Paramtrs Slid 1 Exampl (3 of 3) What is th total rang of put impdancs a sourc could s? A A mpdanc Transformation & Paramtrs Slid 11

12 8/1/18 Spcial Cass of Tratd Transmission s mpdanc Transformation & Paramtrs Slid 3 Shortd, = Rflction from oad 1 and oltag Standg Wav Ratio SWR Thr xists som whr () =. and nput mpdanc j tanh lossy tan losslss Not 1: for th losslss l is purly imagary. This mans it is purly ractiv and no dissipation occurs th l. Th put impdanc altrnats btwn bg capacitiv and ductiv as you back away from th load. [ ] and [ ] short circuit opn circuit Not : Th shortd l bhavs much th sam way as th opn circuit l. W also obsrv that,short,opn mpdanc Transformation & Paramtrs Slid 4 1

13 8/1/18 Opn Circuit, = Rflction from oad 1 and oltag Standg Wav Ratio SWR Thr xists som whr () =. and nput mpdanc coth lossy j cot losslss Not 1: for th losslss l is purly imagary. This mans it is purly ractiv and no dissipation occurs th l. Th put impdanc altrnats btwn bg capacitiv and ductiv as you back away from th load. [ ] and [ ] short circuit opn circuit Not : Th opn circuit l bhavs much th sam way as th shortd l. W also obsrv that,short,opn mpdanc Transformation & Paramtrs Slid 5 Matchd, = Rflction from oad oltag Standg Wav Ratio SWR 1 bcaus = and and nput mpdanc [ ] and [ ] Not: For th matchd l, thr ar no rflctions and all of th powr is dlivrd to th load. mpdanc Transformation & Paramtrs Slid 6 13

Lecture Outline. Shorted line (Z L = 0) Open circuit line (Z L = ) Matched line (Z L = Z 0 ) 9/28/2017. EE 4347 Applied Electromagnetics.

Lecture Outline. Shorted line (Z L = 0) Open circuit line (Z L = ) Matched line (Z L = Z 0 ) 9/28/2017. EE 4347 Applied Electromagnetics. 9/8/17 Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagnetics Topic 4b Transmission ine Behavior Transmission These ine notes

More information

Lecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e

Lecture Outline. Skin Depth Power Flow 8/7/2018. EE 4347 Applied Electromagnetics. Topic 3e 8/7/018 Cours Instructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 3 Skin Dpth & Powr Flow Skin Dpth Ths & Powr nots Flow may contain

More information

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)

More information

Definition1: The ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions.

Definition1: The ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions. Dirctivity or Dirctiv Gain. 1 Dfinition1: Dirctivity Th ratio of th radiation intnsity in a givn dirction from th antnna to th radiation intnsity avragd ovr all dirctions. Dfinition2: Th avg U is obtaind

More information

CHAPTER 10. Consider the transmission lines for voltage and current as developed in Chapter 9 from the distributed equivalent circuit shown below.

CHAPTER 10. Consider the transmission lines for voltage and current as developed in Chapter 9 from the distributed equivalent circuit shown below. CHAPTER 1 1. Sinusoidal Stady Stat in Transmission ins 1.1 Phasor Rprsntation of olta and Currnt Wavs Considr th transmission lins for volta and currnt as dvlopd in Chaptr 9 from th distributd quivalnt

More information

2/12/2013. Overview. 12-Power Transmission Text: Conservation of Complex Power. Introduction. Power Transmission-Short Line

2/12/2013. Overview. 12-Power Transmission Text: Conservation of Complex Power. Introduction. Power Transmission-Short Line //03 Ovrviw -owr Transmission Txt: 4.6-4.0 ECEGR 45 owr ystms Consrvation of Complx owr hort in owr Transmission owr Transmission isualization Radial in Mdium and ong in owr Transmission oltag Collaps

More information

ECE 344 Microwave Fundamentals

ECE 344 Microwave Fundamentals ECE 44 Microwav Fundamntals Lctur 08: Powr Dividrs and Couplrs Part Prpard By Dr. hrif Hkal 4/0/08 Microwav Dvics 4/0/08 Microwav Dvics 4/0/08 Powr Dividrs and Couplrs Powr dividrs, combinrs and dirctional

More information

EIE 332 Electromagnetics

EIE 332 Electromagnetics EIE 332 Elctromagntics cturr: Dr. W.Y.Tam Room no.: DE604 Phon no.: 27666265 -mail: nwytam@polyu.du.hk wb: www.n.polyu.du.hk/~m/mypag.htm Normal Offic hour: 9:00am 5:30pm (Mon-Fri) 9:00am 12:30pm (Sat)

More information

ECE 2210 / 00 Phasor Examples

ECE 2210 / 00 Phasor Examples EE 0 / 00 Phasor Exampls. Add th sinusoidal voltags v ( t ) 4.5. cos( t 30. and v ( t ) 3.. cos( t 5. v ( t) using phasor notation, draw a phasor diagram of th thr phasors, thn convrt back to tim domain

More information

Exercise 1. Sketch the graph of the following function. (x 2

Exercise 1. Sketch the graph of the following function. (x 2 Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability

More information

The pn junction: 2 Current vs Voltage (IV) characteristics

The pn junction: 2 Current vs Voltage (IV) characteristics Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n

More information

General Notes About 2007 AP Physics Scoring Guidelines

General Notes About 2007 AP Physics Scoring Guidelines AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation

More information

NARAYANA I I T / P M T A C A D E M Y. C o m m o n P r a c t i c e T e s t 1 6 XII STD BATCHES [CF] Date: PHYSIS HEMISTRY MTHEMTIS

NARAYANA I I T / P M T A C A D E M Y. C o m m o n P r a c t i c e T e s t 1 6 XII STD BATCHES [CF] Date: PHYSIS HEMISTRY MTHEMTIS . (D). (A). (D). (D) 5. (B) 6. (A) 7. (A) 8. (A) 9. (B). (A). (D). (B). (B). (C) 5. (D) NARAYANA I I T / P M T A C A D E M Y C o m m o n P r a c t i c T s t 6 XII STD BATCHES [CF] Dat: 8.8.6 ANSWER PHYSIS

More information

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.

COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH. C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH

More information

That is, we start with a general matrix: And end with a simpler matrix:

That is, we start with a general matrix: And end with a simpler matrix: DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss

More information

Hyperbolic Functions Mixed Exercise 6

Hyperbolic Functions Mixed Exercise 6 Hyprbolic Functions Mid Ercis 6 a b c ln ln sinh(ln ) ln ln + cosh(ln ) + ln tanh ln ln + ( 6 ) ( + ) 6 7 ln ln ln,and ln ln ln,and ln ln 6 artanh artanhy + + y ln ln y + y ln + y + y y ln + y y + y y

More information

PHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS

PHYSICS 489/1489 LECTURE 7: QUANTUM ELECTRODYNAMICS PHYSICS 489/489 LECTURE 7: QUANTUM ELECTRODYNAMICS REMINDER Problm st du today 700 in Box F TODAY: W invstigatd th Dirac quation it dscribs a rlativistic spin /2 particl implis th xistnc of antiparticl

More information

Section 11.6: Directional Derivatives and the Gradient Vector

Section 11.6: Directional Derivatives and the Gradient Vector Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th

More information

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the

The graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th

More information

5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd

5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd 1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as

More information

A Propagating Wave Packet Group Velocity Dispersion

A Propagating Wave Packet Group Velocity Dispersion Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to

More information

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula

u x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula 7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting

More information

CHAPTER 24 HYPERBOLIC FUNCTIONS

CHAPTER 24 HYPERBOLIC FUNCTIONS EXERCISE 00 Pag 5 CHAPTER HYPERBOLIC FUNCTIONS. Evaluat corrct to significant figurs: (a) sh 0.6 (b) sh.8 0.686, corrct to significant figurs (a) sh 0.6 0.6 0.6 ( ) Altrnativly, using a scintific calculator,

More information

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.

Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J. Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7

More information

Calculus concepts derivatives

Calculus concepts derivatives All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving

More information

Impedance (T) EELE 461/561 Digital System Design. Module #3 Interconnect Modeling with Distributed Elements Topics. Transmission Lines

Impedance (T) EELE 461/561 Digital System Design. Module #3 Interconnect Modeling with Distributed Elements Topics. Transmission Lines EEE 46/56 igital Systm sign Modul #3 ntrconnct Modling with istributd Elmnts Topics. mpdanc of Transmission ins Ttbook Rading Assignmnts Transmission ins mpdanc T - Transmission ins ar istributd lmnts

More information

SAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS

SAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS

More information

A. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).

A. Limits and Horizontal Asymptotes ( ) f x f x. f x. x ±# ( ). A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,

More information

The Frequency Response of a Quarter-Wave Matching Network

The Frequency Response of a Quarter-Wave Matching Network 4/1/29 Th Frquncy Rsons o a Quartr 1/9 Th Frquncy Rsons o a Quartr-Wav Matchg Ntwork Q: You hav onc aga rovidd us with conusg and rhas uslss ormation. Th quartr-wav matchg ntwork has an xact SFG o: a Τ

More information

EXST Regression Techniques Page 1

EXST Regression Techniques Page 1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy

More information

Higher order derivatives

Higher order derivatives Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of

More information

Brief Introduction to Statistical Mechanics

Brief Introduction to Statistical Mechanics Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.

More information

Voltage, Current, Power, Series Resistance, Parallel Resistance, and Diodes

Voltage, Current, Power, Series Resistance, Parallel Resistance, and Diodes Lctur 1. oltag, Currnt, Powr, Sris sistanc, Paralll sistanc, and Diods Whn you start to dal with lctronics thr ar thr main concpts to start with: Nam Symbol Unit oltag volt Currnt ampr Powr W watt oltag

More information

Mor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration

More information

Pipe flow friction, small vs. big pipes

Pipe flow friction, small vs. big pipes Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction

More information

SIGNIFICANCE OF SMITH CHART IN ANTENNA TECHNOLOGY

SIGNIFICANCE OF SMITH CHART IN ANTENNA TECHNOLOGY SIGNIFICANCE OF SMITH CHART IN ANTENNA TECHNOLOGY P. Poornima¹, Santosh Kumar Jha² 1 Associat Profssor, 2 Profssor, ECE Dpt., Sphoorthy Enginring Collg Tlangana, Hyraba (Inia) ABSTRACT This papr prsnts

More information

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches. Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M

More information

Module 8 Non equilibrium Thermodynamics

Module 8 Non equilibrium Thermodynamics Modul 8 Non quilibrium hrmodynamics ctur 8.1 Basic Postulats NON-EQUIIRIBIUM HERMODYNAMICS Stady Stat procsss. (Stationary) Concpt of ocal thrmodynamic qlbm Extnsiv proprty Hat conducting bar dfin proprtis

More information

Chapter 6 Folding. Folding

Chapter 6 Folding. Folding Chaptr 6 Folding Wintr 1 Mokhtar Abolaz Folding Th folding transformation is usd to systmatically dtrmin th control circuits in DSP architctur whr multipl algorithm oprations ar tim-multiplxd to a singl

More information

(1) Then we could wave our hands over this and it would become:

(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 information

Differential Equations

Differential Equations Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations

More information

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic

More information

Quasi-Classical States of the Simple Harmonic Oscillator

Quasi-Classical States of the Simple Harmonic Oscillator Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats

More information

Part 7: Capacitance And Capacitors

Part 7: Capacitance And Capacitors Part 7: apacitanc And apacitors 7. Elctric harg And Elctric Filds onsidr a pair of flat, conducting plats, arrangd paralll to ach othr (as in figur 7.) and sparatd by an insulator, which may simply b air.

More information

Differentiation of Exponential Functions

Differentiation of Exponential Functions Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of

More information

Bifurcation Theory. , a stationary point, depends on the value of α. At certain values

Bifurcation Theory. , a stationary point, depends on the value of α. At certain values Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local

More information

5.80 Small-Molecule Spectroscopy and Dynamics

5.80 Small-Molecule Spectroscopy and Dynamics MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts

More information

There is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.

There is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real. Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual

More information

Text: WMM, Chapter 5. Sections , ,

Text: WMM, Chapter 5. Sections , , Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl

More information

0WAVE PROPAGATION IN MATERIAL SPACE

0WAVE PROPAGATION IN MATERIAL SPACE 0WAVE PROPAGATION IN MATERIAL SPACE All forms of EM nrgy shar thr fundamntal charactristics: 1) thy all tral at high locity 2) In traling, thy assum th proprtis of was 3) Thy radiat outward from a sourc

More information

Theory and Applications of Transmission Lines

Theory and Applications of Transmission Lines Thory and Applications of Transmission ins 1 Introduction Typs Topics Gnral Transmission-in Equations Wav Charactristics on Finit Transmission ins Wavguids Optical Fibr 2 Transmission ins Usd for guiding

More information

ECE602 Exam 1 April 5, You must show ALL of your work for full credit.

ECE602 Exam 1 April 5, You must show ALL of your work for full credit. ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b

More information

Thus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.

Thus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases. Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn

More information

Einstein Equations for Tetrad Fields

Einstein Equations for Tetrad Fields Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for

More information

Electronic Circuits. BJT Amplifiers. Manar Mohaisen Office: F208 Department of EECE

Electronic Circuits. BJT Amplifiers. Manar Mohaisen Office: F208   Department of EECE Elctronic Circuits BJT mplifirs Manar Mohaisn Offic: F208 Email: manar.subhi@kut.ac.kr Dpartmnt of EECE viw of th Prcdnt Lctur Explain th DC Oprating Point Explain th Voltag-dividr Bias Othr Bias Mthods

More information

First derivative analysis

First derivative analysis Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points

More information

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator. Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

Scattering Parameters. Scattering Parameters

Scattering Parameters. Scattering Parameters Motivatio cattrig Paramtrs Difficult to implmt op ad short circuit coditios i high frqucis masurmts du to parasitic s ad Cs Pottial stability problms for activ dvics wh masurd i oopratig coditios Difficult

More information

MSLC Math 151 WI09 Exam 2 Review Solutions

MSLC Math 151 WI09 Exam 2 Review Solutions Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

Synchronous machines

Synchronous machines Synchronous gnrator (altrnator): transorms mchanical nrgy into lctric nrgy; dsignd to gnrat sinusoidal oltags and currnts; usd in most powr plants, or car altrnators, tc. Synchronous motor: transorms lctric

More information

2. Background Material

2. Background Material S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds

More information

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

MATH 1080 Test 2-SOLUTIONS Spring

MATH 1080 Test 2-SOLUTIONS Spring MATH Tst -SOLUTIONS Spring 5. Considr th curv dfind by x = ln( 3y + 7) on th intrval y. a. (5 points) St up but do not simplify or valuat an intgral rprsnting th lngth of th curv on th givn intrval. =

More information

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM

COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,

More information

Lecture 26: Quadrature (90º) Hybrid.

Lecture 26: Quadrature (90º) Hybrid. Whits, EE 48/58 Lctur 26 Pag f Lctur 26: Quadratur (9º) Hybrid. Back in Lctur 23, w bgan ur discussin f dividrs and cuplrs by cnsidring imprtant gnral prprtis f thrand fur-prt ntwrks. This was fllwd by

More information

Waves in cavities such as vehicle compartments, rooms or ducts

Waves in cavities such as vehicle compartments, rooms or ducts 7.1 Wavs in cavitis such as vhicl compartmnts, rooms or ducts Sound propagation from sourcs into th fr fild is rlativly simpl. At a rciving position in a distanc to th sourc th sound will arriv dlayd by

More information

PHYS ,Fall 05, Term Exam #1, Oct., 12, 2005

PHYS ,Fall 05, Term Exam #1, Oct., 12, 2005 PHYS1444-,Fall 5, Trm Exam #1, Oct., 1, 5 Nam: Kys 1. circular ring of charg of raius an a total charg Q lis in th x-y plan with its cntr at th origin. small positiv tst charg q is plac at th origin. What

More information

Recall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1

Recall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1 Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1

More information

ECEN 5004, Spring 2018 Active Microwave Circuits Zoya Popovic, University of Colorado, Boulder LECTURE 2 SOME PASSIVE CIRCUITS

ECEN 5004, Spring 2018 Active Microwave Circuits Zoya Popovic, University of Colorado, Boulder LECTURE 2 SOME PASSIVE CIRCUITS ECEN 54, pring 18 Activ Microwav Circuits Zoya Popovic, Univrsity of Colorado, Bouldr LECTURE OME PAIVE CIRCUIT W hav alrady rviwd atching circuits, which ar -port ntworks. Thy ar passiv and can b rciprocal

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

DSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer

DSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016 LECTURE

More information

Why is a E&M nature of light not sufficient to explain experiments?

Why is a E&M nature of light not sufficient to explain experiments? 1 Th wird world of photons Why is a E&M natur of light not sufficint to xplain xprimnts? Do photons xist? Som quantum proprtis of photons 2 Black body radiation Stfan s law: Enrgy/ ara/ tim = Win s displacmnt

More information

Optics and Non-Linear Optics I Non-linear Optics Tutorial Sheet November 2007

Optics and Non-Linear Optics I Non-linear Optics Tutorial Sheet November 2007 Optics and Non-Linar Optics I - 007 Non-linar Optics Tutorial Sht Novmbr 007 1. An altrnativ xponntial notion somtims usd in NLO is to writ Acos (") # 1 ( Ai" + A * $i" ). By using this notation and substituting

More information

PHA 5127 Answers Homework 2 Fall 2001

PHA 5127 Answers Homework 2 Fall 2001 PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring

More information

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.

Note If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark. . (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold

More information

1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by:

1.2 Faraday s law A changing magnetic field induces an electric field. Their relation is given by: Elctromagntic Induction. Lorntz forc on moving charg Point charg moving at vlocity v, F qv B () For a sction of lctric currnt I in a thin wir dl is Idl, th forc is df Idl B () Elctromotiv forc f s any

More information

Lecture 3: Phasor notation, Transfer Functions. Context

Lecture 3: Phasor notation, Transfer Functions. Context EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of

More information

The Transmission Line Wave Equation

The Transmission Line Wave Equation 1//5 Th Transmission Lin Wav Equation.doc 1/6 Th Transmission Lin Wav Equation Q: So, what functions I (z) and V (z) do satisfy both tlgraphr s quations?? A: To mak this asir, w will combin th tlgraphr

More information

Sundials and Linear Algebra

Sundials and Linear Algebra Sundials and Linar Algbra M. Scot Swan July 2, 25 Most txts on crating sundials ar dirctd towards thos who ar solly intrstd in making and using sundials and usually assums minimal mathmatical background.

More information

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields

Lecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration

More information

Abstract Interpretation: concrete and abstract semantics

Abstract Interpretation: concrete and abstract semantics Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion

More information

Coupled Pendulums. Two normal modes.

Coupled Pendulums. Two normal modes. Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron

More information

Content Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles.

Content Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles. Tachr: CORE TRIGONOMETRY Yar: 2012-13 Cours: TRIGONOMETRY Month: All Months S p t m b r Angls Essntial Qustions Can I idntify draw ngativ positiv angls in stard position? Do I hav a working knowldg of

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1

More information

MCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems

MCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a

More information

September 23, Honors Chem Atomic structure.notebook. Atomic Structure

September 23, Honors Chem Atomic structure.notebook. Atomic Structure Atomic Structur Topics covrd Atomic structur Subatomic particls Atomic numbr Mass numbr Charg Cations Anions Isotops Avrag atomic mass Practic qustions atomic structur Sp 27 8:16 PM 1 Powr Standards/ Larning

More information

DIELECTRIC AND MAGNETIC PROPERTIES OF MATERIALS

DIELECTRIC AND MAGNETIC PROPERTIES OF MATERIALS DILCTRIC AD MAGTIC PROPRTIS OF MATRIALS Dilctric Proprtis: Dilctric matrial Dilctric constant Polarization of dilctric matrials, Typs of Polarization (Polarizability). quation of intrnal filds in liquid

More information

Chapter 6: Polarization and Crystal Optics

Chapter 6: Polarization and Crystal Optics Chaptr 6: Polarization and Crystal Optics * P6-1. Cascadd Wav Rtardrs. Show that two cascadd quartr-wav rtardrs with paralll fast axs ar quivalnt to a half-wav rtardr. What is th rsult if th fast axs ar

More information

PHYS-333: Problem set #2 Solutions

PHYS-333: Problem set #2 Solutions PHYS-333: Problm st #2 Solutions Vrsion of March 5, 2016. 1. Visual binary 15 points): Ovr a priod of 10 yars, two stars sparatd by an angl of 1 arcsc ar obsrvd to mov through a full circl about a point

More information

1973 AP Calculus AB: Section I

1973 AP Calculus AB: Section I 97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th

More information

Alpha and beta decay equation practice

Alpha and beta decay equation practice Alpha and bta dcay quation practic Introduction Alpha and bta particls may b rprsntd in quations in svral diffrnt ways. Diffrnt xam boards hav thir own prfrnc. For xampl: Alpha Bta α β alpha bta Dspit

More information

1 General boundary conditions in diffusion

1 General boundary conditions in diffusion Gnral boundary conditions in diffusion Πρόβλημα 4.8 : Δίνεται μονοδιάτατη πλάκα πάχους, που το ένα άκρο της κρατιέται ε θερμοκραία T t και το άλλο ε θερμοκραία T 2 t. Αν η αρχική θερμοκραία της πλάκας

More information

15. Stress-Strain behavior of soils

15. Stress-Strain behavior of soils 15. Strss-Strain bhavior of soils Sand bhavior Usually shard undr draind conditions (rlativly high prmability mans xcss por prssurs ar not gnratd). Paramtrs govrning sand bhaviour is: Rlativ dnsity Effctiv

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is

More information

Classical Magnetic Dipole

Classical Magnetic Dipole Lctur 18 1 Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg

More information