Solving ODEs with KCL: Circuit Analysis and Complex Numbers
School
University of Illinois, Urbana Champaign**We aren't endorsed by this school
Course
ECE 205
Subject
Electrical Engineering
Date
Dec 11, 2024
Pages
2
Uploaded by UltraResolve11737
ECE205-HW7WrittenName:UIN:1. Using KCL and thev-irelations for resistors and capacitors, show that the voltagev(t)in thefollowing circuit satisfies the following ODE ifR= 5⌦andC= 2F.Also find the capacitorvoltagev(t)ifis(t) =cos(t). Assume zero initial conditions.2dvdt+15v(t) =is(t)fort >0.is(t)RC+v(t)ShivanshAgarwal669/6219Dividingtheequationby2k+fV(t):list)ODweknowthatis(t):cost(E+Y0)V(t)=(cos(t)~solvingtheaboveequation,vi(t)=ce-0.st,thisisthehomogeneoussolution->solvingforparticular,Up(t):Acost+13sint3substitutinginDsAsint+Blost+%(Alost+Bsint)=↓costVp'(t)=- Asint+Blost>!l3-A:O>↓A+B=0cfromand,wegetA=004a5B:0.1993-usingthisand&,C=-00495:V(t):-004960:0.+00495cos(t)+0496sin(t)
2. Find the magnitude and phase of the following complex numbers.Also plot the magnitude and phaseas a function ofomega(a)z =11 + j!, where!is a real number.(b)z =-11 + j!, where!is a real number.usingconjugate,↓-ju(1+jw)(1-ja)=an*sinceju=-..2=Tra-8wrX0:FamilW/tuh=>O=tan"(-w,Yw2/1121/=Huhphot:⑳1+w/inn⑳z=71 (1-54=thisFant(int)1+C3/=>1/2/1=Itwe0:FunWith==tan"(2)Itwh/-theX/