EECE 301 Signls & Systms Pf. Mk Fwl Discussin #1 Cmplx Numbs nd Cmplx-Vlud Functins Rding Assignmnt: Appndix A f Kmn nd Hck
Cmplx Numbs Cmplx numbs is s ts f plynmils. Dfinitin f imginy # j nd sm sulting pptis: j ( ( 1 j)( j) 1 j)( j 1 j) 1 Rcll tht th slutin f diffntil qutins invlvs finding ts f th chctistic plynmil S Rctngul fm f cmplx numb: jb l numbs R{ } b Im{ } Th uls f dditin nd multiplictin stight-fwd: Add : ( Multiply : ( jb) ( c jb)( c jd) ( jd) ( c c) j( b bd) j( d d) bc)
Pl Fm j > 0 Pl fm n ltnt wy t xpss cmplx numb Pl Fm gd f multiplictin nd divisin Nt: yu my hv lnd pl fm s w will NOT us tht h!! Th dvntg f th j is tht whn it is mnipultd using uls f xpnntils nd it bhvs pply ccding t th uls f cmplx #s: ( x )( y ) x y x / y x y Multiplying Using Pl Fm ( j )( ) 1 j j( 1 ) n 1 ( j ) n 1/ n 1/ n j / n n 1 jn Dividing Using Pl Fm ( j ) 1 ( j ) 1 1 j( 1 ) 1 1 1 j j
W nd t b bl cnvt btwn Rctngul nd Pl Fms this is md sy nd bvius by lking t th gmty (nd tignmty) f cmplx #s: Gmty f Cmplx Numbs b Im jb R b b Cnvsin Fmuls sin cs tn 1 b b
Q: Why th fm j f pl fm?? Stt with tignmty: [ cs j sin ] jb cs j( sin ) 4 6 Fm Clc II: cs 1...! 4! 6! j sin j 3 j 3! 5 j 5! 7 j 7!... cs j sin 1 j! 3 j 3! 4 4!... Als Fm Clc II: x 1 x x! 3 x 3! 4 x 4!... j 1 j! 3 j 3! 4 4!... Sinc cs jsin hs th sm xpnsin s j w cn sy tht: cs j sin j
Cmplx Expnntils vs. Sins nd Csins Eul s Equtins: (A) (B) (C) (D)
Summy f Rctngul & Pl Fms Rct Fm: jb R{ } cs Im{ } b sin Pl Fm: j tn b 1 0 b ( π, π ] Wning: Yu clcult will giv yu th wng nsw whnv yu hv < 0. In th wds, f vlus tht li in th II nd III qudnts. Yu cn lwys fix this by ith dding subtcting π. Us cmmn sns lking t th signs f nd b will tll yu wht qudnt is in mk su yu ngl gs with tht!!!
Cnjugt f Z Dntd s * jb * jb j * j Pptis f * 1. * R{ }. * ( jb)( jb) b
Unit Cicl 1 Im R Unit Cicl: A st f cmplx numbs with mgnitud f 1 ( 1) All n th unit cicl lk lik: j 1 Fu spcil pints n th unit cicl: jπ 1 Im jπ j j 1 jπ j0 R ± jnπ ± jnπ jnπ Knw ths!!! 1, n dd intg 1, n vn intg 1 f ll intgs j, n 1,5,9,... j, n 3,7,11,... 1, n 0, 4,8,... 1, n,6,10,...
A sinusid is cmpltly dfind by its th pmts: -Amplitud A (f EE s typiclly in vlts mps th physicl unit) -Fquncy ω in dins p scnd -Phs shift φ in dins T is th pid f th sinusid nd is ltd t th fquncy
Fquncy cn b xpssd in tw cmmn units: -Cyclic fquncy: f 1/T in H (1 H 1 cycl/scnd) -Rdin Fquncy: ω π/t (in dins/scnd) Fm this w cn s tht ths tw fquncy units hv simpl cnvsin fct ltinship (lik ll th unit cnvsins.g. ft nd mts): ω πf Phs shift (ftn just shtnd t phs) shws up xplicitly in th qutin but shws up in th plt s tim shift (bcus th plt is functin f tim). Q: Wht is th ltinship btwn th plt-bsvd tim shift nd th qutinspcifid phs shift? A: W cn wit th tim shift f functin by plcing t by t t (m n this lt, but yu shuld b bl t vify tht this is tu!) Thn w gt: f ( t t) Asin( ω ( t t)) Asin( ωt ωt) S w gt tht: φ ωt (unit-wis this mks sns!!!) φ
In cicuits yu usd phss (w ll cll thm sttic phss h) th pint f using thm is t mk it EASY t nly cicuits tht divn by singl sinusid. H is n xmpl t fsh yu mmy!! Find utput vltg f th fllwing cicuit: R 1Ω ( ) x( t) 5cs 1000t π 4 L mh y( t)? Us phs nd impdnc ids: Impdnc f Induct : Phs f Input : xˆ 5 Z L jπ 4 jωl Us vltg divid t find utput: yˆ j xˆ 1 j 5 jπ [ ] j0.46 0. 4 89 j Output phs: y ˆ 4.45 Output signl: y( t) 4.45cs(1000t 1.5) j1.5
Nt tht in using sttic phss th ws n nd t cy und th fquncy it gts suppssd in th sttic phs BUT if yu hv multipl diving sinusids (ch t its wn uniqu fquncy) thn yu ll nd t kp tht fquncy in th phs psnttin tht lds t: Rtting Phss Kping th fquncy pt A jω t R cs( ω t cs( ω t) j sin( ω t) { jφ jω t} A A cs( ω t φ ) input tting phs φ ) A cs( ω t φ ) A A A sttic phs pt jφ j( ω t φ ) jφ jω t jω t systm Mdifid systm utput tting phs Rtting phs
If ~ x ( t) [ ω t φ ] ( x( t) Acs( ω t φ )) Wht is : A ~ x *( t) [ ω t φ ] jφ jωt A ~ x ( t) ~ x *( t) R{ ~ x ( t) } Acs( ω t j A j φ ) Bcus tting phss tk th vlu f cmplx numb t ch Instnt f tim thy must fllw ll th uls f cmplx numbs Espcilly: EULER S EQUATIONS!!
Rtting Phss Eul s Equtins
Viwing tting phs n th cmplx pln