In the circuit to the right R 125 and C 080 F The switch h

In the circuit to the right, R = 125 and C = 0.80 F. The switch has been open for a long time and is closed at t = 0. If V_s = 20.0V (constant), use LaPlace transforms to find V(S) and v(t) for t > 0. If Vs = 20.0 cos(5,000t)V, use LaPlace transforms to find V(S) and v(t) for t > 0.

Solution

5)

Vs = IR + Vc

=> Vs = RC d Vc/dt + Vc

taking laplace transform on both sides

=> Vs/s = RC( sVc(s) - V0 ) + Vc(s)[ V0 = 0 as initial voltage of capacitor before switch is closed]

=> Vc(s) = Vs( s ( s+ 1/RC))

=> Vc(s) = Vs/s + Vs/( s+ 1/RC)=> Vc(s) = 20/s + 20/( s + 1/10)

=> Vc(t) =Va u(t) - Va e^-t/RC u(t)

=> Vc(t) = 20 u(t) - 20 e^-t/10 u (t)

(b)

Vs = IR + Vc

=> Vs = RC d Vc/dt + Vc

taking laplace transform on both sides

=> Vs/(s² + 25 *10⁶) = 125 * 0.8 ( sVc(s) - V0 ) + Vc(s)[ V0 = 0 as initial voltage of capacitor before switch is closed]

Vs/(s² + 25 *10⁶) = 10 Vc(s) ( 1+ s/10)

Vc(s) = Vs /[10 * (s² + 25 *10⁶) *(1+ s/10)]

vc = L^-1{ Vs /[10 * (s² + 25 *10⁶) *(1+ s/10)]}