Question

Please help solve, providing a detailed solution using the equations provided below and LaPlace transform (Use the table provided in the link) to solve the differential equations obtained when working through the question.

Being given the following information, use the equations provided to find the steady-state current in the following RLC circuit. R=82 L= 0.5H C= 0.1F E(t) = 100 cos(2t) V knowing that at t = 0, i(0) = 0 Equations: UR = Ri VL = = L- di 9 Uci dt С VR + V1 + Vc = e(t) or =V (if the source voltage is constant) dq duc i= = C- q=ſidt dt dt - 9 di L + Ri+ dt = e(t) di L+Ri + Vc = e(t) dt dvc, Considering that, i = C- dt by substitution, we obtain the following equation with which we will be working d²vc duc LC dt2 +RC + Vc = e(t) dt

Link to the Laplace Transform Table:

https://ibb.co/TkrvbNH

Answer 1

2 Lcd²vc + RC a vc + Vc = elt! dt dt² R=8, c=01, L=ois, Elt) = 100 cosat v Viola Vo=0 + 0.05 d²Vc dt² 0.8 dve + c = 100 coszt at using Laplace Tous foom Vc Y YCS) [s? Yes) - s4CO) - V.0] +0-8sY) - Ves] + Ys) loo S 92 + 4 $2+4 (o. oss? 52 +0.8 s + 1) YS) = 100 Y(S) = 100 s (3²+ 4) coross² +0.85+1) Y(S) = 2000S (5²+4 ) ( S²+ 165+20) Y(S) = 20005 (8²74) (5²+165 +20) UJTE Valt) = (275 e" hit + 150 stie +15 OJT +275) 842514 22 t sosinat +25cosat coco

For steady state as to 50 sinat +25 cosat Vc (t) ay to stendy State Cuosent Ć (t) as tto c d Velt dt astro ist) 0.1 [100 cost 100 Cosat - 50 sinat - 50 sinat 5 sinat is (t) = 10 Cosat An