Find v1t and v2t in the following circuit SolutionW 20 rads

Find v1(t) and v2(t) in the following circuit.

W= 20 rad/s

E1=5cos(20t), E2= 10 cos(20t-45)

I= cos(20t+45)

Xc= impedance of capacitor= 1/(jwC)= -0.5j

XL = inductor impedance= jwL= 2j

Apply kCL at node V2.

(E1-E2-V2)/(10+XL) +I = V2/Xc

(E1-E2)/(10+2j). +I= V2[2j+1/(10+2j)]

[-2cos(20t)-7sin(20t)]/(10+2j) +I = V2[-3+20j]/(10+2j)

[3.65 cos(20t)-15.5sin(20t)]= V2(-3+20j)

V2= 15.9cos(20t+77)/(-3+20j)

V2= 20.22cos(20t-22)

Ic= current through capacitor= V2/Xc=40.44 cos(20t+68)

I1= current through 10 ohm= Ic-I

I1= -14.44cos(20t)+36.78sin(20t)

V1= E1-10*I1 = 149.44cos(20t)-367.8sin(20t)