Question

Find the steady state charge and the steady state current in an LRC circuit when L = 1 h, R= 2 Ω , C= 0.25f and E(t)= 50 cost V.

Short answer

The steady state charge and the steady state current in an LRC circuit,

i(t) = q'(t) = 50/13 (2 cost-3 sint) A

Step-by-step solution

Oscillation of spring-mass system:

As in the case of forced oscillations of a spring-mass system with damping, you can call Qp the steady state charge on the capacitor of the LRC circuit. Since I=Q′=Q′_c+Q′_p and Q′_c also tends to zero exponentially as t→∞, you say that I_c=Q′_c is the transient current and I_p=Q′_p is the steady state current.

Finding the steady state charge and the steady state current in an LRC circuit:

In the LRC series electric circuits, you have

L (d²q/dt²)+R (dq/dt)+ q/C= E(t)

(d²q/dt²)+2 (dq/dt)+ 4q= 50 cost

The characteristics equation is given by:

m²+2m+4=0

Whose a repeated root -1+√3i. Thus the complementary solution is:

q_c(t)=e^-t [c₁ cos√3 t+c₂ sin√3t]

Assume that the particular solution is q_p(t)= A cost+ B sint and substituting A= 150/13 and B=100/13,

q_c(t)=50/13 [3 cost+c₂ sint]

Therefore, the steady state current is given by :

i_p(t) = q'_p(t)=50/13 [3 cost+c₂ sint] A

This is the result.