LC Circuit 1 1 2 3 1 1 6 A circuit is constructed with two capacitors and an ind

Question

LC Circuit 1

1

2

3

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1

6

A circuit is constructed with two capacitors and an inductor as shown. The values for the capacitors are: C1 = 405 F and C2 = 190 F. The inductance is L = 207 mH. At time t =0, the current through the inductor has its maximum value IL(0) = 88 mA and it has the direction shown.

1)

What is o, the resonant frequency of this circuit?

radians/s

2)

What is Q1(t1), the charge on the capacitor C1 at time t = t1 = 19 ms? The sign of Q1 is defined to be the sameas the sign of the potential difference Vab = Va - Vb at time t = t1.

C

3)

What is Vbc(t1)= Vb - Vc, the voltage across the inductor at time t1 = 19 ms? Note that this voltage is a signed number.

V

4)

What is Q1,max, the magnitude of the maximum charge on capacitor C1?

C

Explanation / Answer

given

C1 = 405 F

and C2 = 190 F

The inductance is L = 207 mH

At time t =0, the current through the inductor has its maximum value IL(0) = 88 mA

1 )

the resonant frequency of this circuit o   

C = ( 1 / C1 + 1 / C2 )-1

= ( 1/ 405 + 1/190 )-1

= 133.59 uF

o = 1 / ( LC )1/2

o = 1 / ( 207 X 10-3 X 133.59 X 10-6 )1/2

= 190.163 rad/sec

2 )

t = t1 = 19 ms = 0.019 sec

Qmax = IL(0) / o

= 88 X 10-3 / 190.163

= 4.627 X 10-4 C

Q (t1) = Qmax X sinot1

= 4.627 X 10-4 X sin( 190.163 X 0.019 )

= - 2.915 X 10-5 C

3 )

I = 88 X 10-3 coso t

Vbc = - L o Imax sinot

= - 207 X 10-3 X 190.163 X 88 X10-3 X sin(240 X 0.019 X 10-3 )

= 2.75 X 10-4 V

4 )

Q (t1) = Qmax X sinot1

= 4.627 X 10-4 X sin( 190.163 X 0.019 )

= - 2.915 X 10-5 C

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