Announce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review
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1 ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG Appndix A: Complx umbrs Appndix B: MATLAB Chaptr : Sinusoids Chaptr 3: Spctrum 3 LECTURE OBJECTIVES Complx umbr Rviw Conjugat, multiply, powrs, roots of unity Complx Exponntials Rotating phasor, rprsntation of sinusoid by a complx amplitud Adding Sinusoids Sam frquncy: phasor addition thorm Diffrnt frquncis: Th spctrum 1
2 Pop Qui: Find th Roots (a) of th polynomial 1? (b) of th polynomial + 1? COMPLEX UMBERS j 1 COMPLEX ADDITIO = VECTOR Addition A complx numbr is = x + jy: y Im x R Cartsian coordinat systm Som polynomials hav no ral roots (.g. + 1), but all ordr- polynomials hav (possibly complx) roots. 3 1 (4 j3) ( j5) (4 ) j( 3 5) 6 j
3 POLAR Coordinats Eulr: j cos( ) j sin( ) Simpl chang of coordinats: Lngth = r r x y Angl = tan( ) y / x Unxpctd xponntial notation: = x + jy = r j r y = rsin x = rcos Complx Exponntial Ral part is cosin Imaginary part is sin Magnitud is on r j r cos( ) jr sin( ) 9 Most Bautiful Math Formula? Common Valus Common Valus 1 1 j0 j j j0 1 1 j0 j jn j/ j( n1/ ) j( n1) j3/ j/ j( n1/ ) 1 j j/ 4 1 j? 3
4 COMPLEX COJUGATE (*) Usful concpt: chang th sign of all j s RECTAGULAR: If = x + jy, thn th complx conjugat is * = x jy POLAR: Magnitud is th sam but angl has sign chang COMPLEX COJUGATIO Flips vctor about th ral axis! j r * r j USES OF COJUGATIO Z DRILL (Complx Arith) Conjugats usful for many calculations Ral part: * ( x jy) ( x jy) x Imaginary part: * j jy j y 4
5 Invrs Eulr Ral part: Imaginary part: * j j * y j j j j cos( ) j j j sin( ) j Mag & Magnitud Squard Magnitud Squard (polar form): * ( r j )( r j ) r Magnitud Squard (Cartsian form): * ( x jy)( x jy) x Magnitud of complx xponntial is on: j cos j sin 1 y x y COMPLEX MULTIPLY = VECTOR ROTATIO Multiplication/division scals and rotats vctors POWERS = 1 r j r j 5
6 MORE POWERS BOTTOM LIE < 1 > 1 CARTESIA: Addition/subtraction is most fficint in Cartsian form POLAR: good for multiplication/division STEPS: Idntify arithmtic opration Convrt to asy form Calculat Convrt back to original form ROOTS OF UITY How many solutions to? r j 1 jk k r 1, k jk, k 0, 1,, 1 1 ROOTS OF UITY for =6 1 Solutions to ar qually spacd vctors on th unit circl! What happns if w tak th sum of all of thm? 6
7 Sum th Roots of Unity Looks lik th answr is ro (for = 6) 1 k0 jk / Writ as gomtric sum 0 always? Pop Qui Find valus of for which: ( -5) = 0. 1 k0 k 1 thn lt 1 j/ umrator 1 1 ( ) j/ 1 j 0. HISTORY Adding Sinusoids via Phasor Addition Which company s first succssful product was a sin-wav gnrator? Variabl frquncy Lab Instrumnt 7
8 Rviw of COMPLEX EXPOETIAL A sinusoid is th ral part of a complx xponntial: j4t cos( 4t) j sin( 4t) Complx Amplitud Rwrit a gnral sinusoid lik this: x( t) A cos( t ) R Whr th complx amplitud is X X j t A j Th Complx Amplitud A 400-H sinusoid lik this: x( t) 3. cos( 800t 0. 3), whn th frquncy 400 H is undrstood, is convnintly rprsntd by its complx j0. 3 amplitud X 3., according to: Givn 400-H, X tlls you vrything. x( t) R X j 800 t POP QUIZ: Complx Amp Find th COMPLEX AMPLITUDE for: x( t) 3 cos( 77t 0. 5) Us EULER s FORMULA: x( t) X 3 3 j 0. 5 j 77 t j0. 5 8
9 POP QUIZ-: Complx Amp Dtrmin th 60-H sinusoid whos COMPLEX AMPLITUDE is: Convrt X to POLAR: x( t) ( 1 X j / 3 3 j3) 3 j3 j( 10 t) j 10 t Motivating Qustion: How to Add Sinusoids? What is: 1. 1cos( 70t 0. 1) 0. 8 cos( 70t 0. 6)? x( t) 1 cos( 10t / 3) 1. 1cos( 70t 0. 1) 1. 1cos( 70t 0. 1) 0. 8 cos( 70t 0. 6) 0. 8 cos( 70t 0. 6) It looks sinusoidal! 5/1/018 ECE-05 Spring-005 JMc 35 5/1/018 ECE-05 Spring-005 JMc 36 9
10 Ky Rsult Why? Phasors Maks it Clar: Th sum of sinusoids having th sam frquncy is a singl sinusoid of that frquncy! (Rgardlss of amplituds and phass.) 1. 1cos( 70t 0. 1) 0. 8 cos( 70t 0. 6) ECE-05 Spring-005 JMc PHASOR ADDITIO RULE A cos( t ) A cos( k 1 k 0 k 0 t ) whr A and ar found by adding th phasors of th sinusoids: j k A 1 A k j k 10
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