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1 EE334 - Wavs and Phasos Lcu: pp This cous aks vyhing ha you hav bn augh in physics, mah and cicuis and uss i. Easy, only nd o know 4 quaions: 4 wks on fou quaions D ρ Gauss's Law B No Monopols B E Faaday's Law D H J mp's Law W will div all h cicui pssions: Ohm s Law (laion), KCL, KL, L, R, C fom Mawll s quaions and oh fundamnal laionships. W will simplify hm by assuming no im dpndnc saic cass W will combin hm ino a scond od diffnial quaion wav quaion Bu fis nd o dvlop h mahmaics quid o alk abou EM Wha is a scala? Quaniy wih magniud: nso of o ank E: im, mass, disanc, mpau, spd I was only going 75 mph Scala quaion? IR Wha is a vco? Quaniy wih magniud and dicion: nso of fis ank E: foc, displacmn, lcic fild innsiy, vlociy H was going 95 mph (magniud) as on I-9 (dicion) co quaion: J y E J σ σ σ σ E J y σ y σ yy σ y E y J σ E σ y E y σ E J σ σ y σ E J σ E assums a summaion of pad indics (Einsin s noaion) i i -

2 Th cun dnsiy is qual o h conduciviy ims h lcic fild Th conduciviy is a maial popy ha can b a funcion of dicion (as in cysals) no h sam in all dicions anisoopic I is a nso of scond ank: h numb of subscips is h ank of h nso. Wha is a fild? of posiion. Funcion ha spcifis a quaniy vywh in a gion: funcion E: cclom in my ca as I dov fom Boman o Billings on I-9. dv d a Inga onc givs m h vlociy of my vhicl, d d wha is h dicion? (dicion ca is hading nd 6 snsos o complly dfin inial spac) Inga wic givs m h displacmn, Wha is? Th pah ha my ca aks, lin ingal. Saw MHP Bllgs BZ Lngh of aow psns magniud Passd cal uck Wha is flu? No h suff you us whn you sold Masu a vco hough a sufac look a h ciy limis of Boman o b h donu. Can b modld as a paial diffnial quaion -

3 Th donu o Boman Ciy limis N flu of poplabou if > fillup wih popl if< vy on lf Don g los in h mah, us a psnaion of h physical wold (symbols) 5 w all know wha i mans, how abou if I add anoh symbol $5 How abou his on? pac, vicoy, l, caa ins, oman numal 5? Pong gam, CI, I O I, bas (5) EM wav adia fom souc, if you a fa nough away h cuvd wav fons look plana. Plan wav in a losslss mdium: h wav dos no anua h ampliud as h wav avls has h fom: y (, ) T λ : ampliud T: piod λ: spaial wavlngh : fnc phas Can also wi as: -3

4 y (, ) [ (, ) ] T λ (, ) Th angl is calld h phas of h wav and is a funcion of posiion and im Boh im and posiion ca sinusoidal ampliud whn h oh is fid. Th paks and oughs mov wih a popagaion vlociy, fo a givn ampliud h phas is a consan: C T λ d d Taking h divaiv of h quaion givs h phas vlociy T λ d λ u p d T -4

5 Th fquncy of h sinusoidal wav is h cipocal of h piod: f T nd h phas vlociy is lad o h fquncy and wavlngh: u p fλ Th quaion fo h ampliud as a funcion of posiion and im can b win in h fom: y λ λ Phas consan o wav numb (, ) f ( ) f ngula Fq. u p Can ins h ab. Phas shif: y (, ) ( ) -5

6 Posiiv phas shif lads fnc wav, ngaiv phas shif lags fnc wav. In a lossy mdium h ampliud is anuad nuaion faco: nuaion cof: α α Gnal fom of a sinusoidal ampliud wih anuaion: α y, Compl Numbs Rviw Can wi a compl numb as a al pa and an imaginay pa: y. R {} Im y Imaginay is y (,y)!! θ Ral is -6

7 This can b plod in h Casian plan and convd o a pola fom of a magniud and a phas angl: θ θ sho hand noaion usful in calculaions Can us Eul s Indniy: θ θ sinθ θ θ θ y sinθ y y sinθ θ an y Rflcion as h al ais givs h compl conuga ( plac wih ) y θ Th magniud of a compl numb can b calculad fom h compl conugas: Compl Mah: ddiion: add al pas and add imaginay pas: ( ) ( y ) y Muliplicaion: ( y y ) ( y y ) ( θ ) θ Division: y y θ θ y y ( y y ) ( y y ) y n θ n n nθ n Pows: ( ) ( nθ sin nθ ) Phasos Rviw: sin R { } ( ) -7

8 -8 Wi in phaso noaion: sin Us sin() (/-): () Us vn funcion (-) (): () Us Phaso noaion () { } R () R { } R R conains h phas and ampliud infomaion bu no h fquncy () sin sin sin α α d d d d d d

DSP-First, 2/e. This Lecture: LECTURE #3 Complex Exponentials & Complex Numbers. Introduce more tools for manipulating complex numbers

DSP-First, 2/e. This Lecture: LECTURE #3 Complex Exponentials & Complex Numbers. Introduce more tools for manipulating complex numbers DSP-Fis, / LECTURE #3 Compl Eponnials & Compl umbs READIG ASSIGMETS This Lcu: Chap, Scs. -3 o -5 Appndi A: Compl umbs Appndi B: MATLAB Lcu: Compl Eponnials Aug 016 003-016, JH McClllan & RW Schaf 3 LECTURE