Question

The following circuit consists of linear time invariant components where R₁ = 2$\Omega$, R₂ = 3$\Omega$, C = 0.25F, and L = 1H. Let v_s be the input voltage.

(a) Solve the differential equation that describes v_c(t). Hint: First, find the current going through the parallel branch of R₁–C. Then, write the Kirchoff's Voltage Law for the main loop.

(b) Find v_c(t) given that v_s(t) = 2.5 V at t = 0. Assume that v_c(0^-) = 1V and i_L(0^-) = 2A.

Answer 1

Solunm-> R1 =25 R=35 L-IH + Vs + Vc (1 (a) Applying KvL in loop ③ Vs (t) = Rp ilt) + Velt) + L dicts dict Vs (t) = 3 i (t) + Vc (t) + et Differential Equation that Describes Velt). (b) Drawing Laplace Transformed circuit. 3 1,199 mm 15 Lico) ILS) 194 - Vs (S) (+ S V.16). tk Ve(s) - S 2.5 S Vs (S) = 3103) + 2(105) - I.()) + SICS) 2.5 +2 - egn (5+5) ICS) - 2 Ic(s)

Page 47 of 76 write this Applying KPL in loop 2(I C5) - Iels)) - 7,05)-} 2105) - 91c(s) - Ic(S) - ICS) - Ic(s) (5+2) - egna S IS 2-5 S S 25 sub.en & in egn (5+5) 1c (S) (5+2) - 21 (CS) = [ S+ 2] CS45) -I (S+5) +2.5+2 1ecs)[ +75+10-25] * + 2.5+25 Ic (S) (8 + 58+10) = IC (S) 2 25 S 3 2 S 3 s 2 (s² + 58+10) >en (3) IC CS) = sub. egn ③ in Vc CS) Egn. Vc (3) = 4 5 [(s) + Velos 2 32+50+10 3 S + S S Ve(s) = (3+2+5)*+/3.5)2

ㅗ S Y(S) = Х 3.5 (5+2.5jet (3.5)2 taking caplane Inverse Transform. Velt) = è 2.5t sin (√3.5 t) 41t+ Ult √3.5 Veld) = [3.20.8 2.5€ sin (1.87*) + 1] HIA) Any