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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 + VL + VR  = 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 = IL = 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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