PLEASE SHOW ALL WORK FOR CREDIT!!! THANK YOU!!! In a subsequent semester, the st

Question

PLEASE SHOW ALL WORK FOR CREDIT!!! THANK YOU!!!

In a subsequent semester, the student studies RLC circuits and uses her knowledge of second order linear differential equations. In particular, she investigates an RLC circuit with parameters: R = 6Ohm L = 3H C = 1/15F E(t) = 51 sin (2t) q(0) = q0 and q'(0) = i(0) = i0 Is the circuit underdamped, overdamped or critically damped? State the IVP that governs q(t), the charge. Find a particular solution to the IVP and explain its significance. Find qc(t), the transient part of the charge. Find the steady-state current in this circuit.

Explanation / Answer

(a). The circuit is underdamped .

(b). Equation of IVP:

Balancing current in the circuit , we can say that

i(t) = C * dE/dt + E/R + L di/dt ............... where symbols have usual meaning .

dE/dt = 102 cos (2t)

Given , R = 6 ; L = 3 ; C = 1/15

Substituting this in the above stated equation, we get.

i(t) = 6.8 cos(2t) + 8.5 sin(2t) + 3 i'

We can write i = dq/dt

Substituting this in above equation, we get

q' = 6.8 cos(2t) + 8.5 sin(2t) + 3 q''

3*q''-q' = - (6.8 cos(2t) + 8.5 sin(2t)) ................ans.

c). Let q = A cos(2t) + B sin(2t)

Put q in the IVP to find A and B

which gives A = 0.68 and B = (8.5)/(-14) = 0.607

Put these values in assumed equation of q.

e) I = 51/Z

Xr = 6

Xl = 6

Xc = 15/2 = 7.5

Z = 6.184

I = 8.25 Amperes.

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