Understanding Complex Numbers and Sinusoidal Integration
School
University of California, Los Angeles**We aren't endorsed by this school
Course
MAE 107
Subject
Electrical Engineering
Date
Dec 11, 2024
Pages
5
Uploaded by ChefRose22801
#Agenda:Reviewofcomplexnumbers·Integrationinvolvingsinusoids1:(1)Plotthefollowingonacomplexplane(2)Computemag.(p)andphase(0)andexpressinexponentialnotation:peso-EtI0=tait)rador450pet(4)(toconvertbacktoRel,Em),useuler'sformulai(En=1+jb)1+jrador139I=pet-te((4)
3)Imeid)1.j--#2:Expressthecomplex#in:(1)Complexnotationx+jB(2)Exponential:pejoa)(1+j)()-j)=j-(-1)=E]-Et#b)ej+j=cos()+ji))jei)cos()()(.#Gcos()=E=GRele*5)
)e-j-)+jsin()=2jsin()-K=reme)#Ex3:Consider'swhereSjc,forw=001,01,1,10,100#is(iii)Plotmag.vs.freg(P4a)Plotphasevsfreg.(Evsa)#=10ej0-98)=Hiitt=E5Mag&..+1)Dj10"I'soisidewradl'ssirt)i108Phase,o&90I15"10%idoisonwrad))
#4:Showthat(elt-sin(we do=sin (w+++)-sintwherep=tari'(w)orsinp=andcostHintilatestejut,wee(t-sinwd-eiwewd-ele·tellite·↑*Expandviafuler'sforaset-)(cos(wt)+jsin(wt)-(0s(wt)-jsin (w+)Het)F-·
Trigidentity:cos(p)+cos(alsin(p)=sin(x+3)wa(sin (w++P)-e+sin(p)#5:ForW30andT=,showthefollowing:a)sin(wood==0&sin(word==i cos(wor)1Integratingsineandcosineover=w(cos(woT)-1)oneperiodyieldszero=-w(cos(2π)-1)-lo)b)(cos(wod=0&&costcod+=sin(wot)(T=w(sin(am)-0)-D