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Part b)

After removing power supply from circuit and taking switch in position 2,

Voltage drop across to capacitor is given by

Vc + V_L + V_R = 0

ð LdI/dt + IR + Vc

ð LCd²Vc/dt² +RCdVc/dt+ Vc =0

The characteristic equation is

LCs² +RCs +1 =0

ð s² +(R/L)s +1/LC =0

ð $s = -\alpha \pm \sqrt{\alpha^2 - \beta^2}$

Where

α = R/2L = (4.7 +1)/(2*330) = 0.0086

β = (1/LC)^1/2 = 1/(1000*10^-6*330*10³)^1/2 =0.055

Hence Vc = [B1 Cos (0.055 t) +B2 Sin(0.055 t)]

At t = 0, Vc = 1.5 V

èB1 = 1.5

Vc = [15 Cos (0.055 t) +B2 Sin(0.055 t)]

Also I = I_L = 0 at t = 0

ð d(CVc)/`dt = 0

ð B2 =1.5*(0.055/0.0086) = 0.64

Vc = [1.5 Cos (0.055 t) +0.64 Sin(0.055 t)]

From the figure it is obvious that system is Under damped

From the Graph of Vc vs time, it is clear that Maximum Vc = 15 at t = 0 and Min Vc = 0 at tàinfinite

Part	B	A
Max Vc	1.5V at t = 0	2.33V at t =75
Min Vc	0 at t à infinite	0 V at t = 0

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