A circuit is constructed with a resistor, two inductors, one capacitor, one batt

Question

A circuit is constructed with a resistor, two inductors, one capacitor, one battery and a switch as shown. The value of the resistance is R1-295 ?. The values for the inductances are: L1-203 mH and L2 = 177 mH . The capacitance is C = 95 ?F and the battery voltage is V- 12 V. The positive terminal of the battery is indicated with a sign 1) The switch has been closed for a long time when at time t - 0, the switch is opened. What is UL1(0), the magnitude of the energy stored in inductor L1 just after the switch is opened? Submit 2) What is , the resonant frequency of the circuit just after the switch is opened? adians/s Submit 3) What is Qmax, the magnitude of the maximum charge on the capacitor after the switch is opened? Submit 4) What is Q(q), the charge on the capacitor at time t-t1-3 ms, Q(t1) is defined to be positive if V(a)-V(b) is positive Submit 5) What is t2, the first time after the switch is opened that the energy stored in the capacitor is a maximum? Submit 6) What is the total energy stored in the inductors plus the capacitor at time t-t2?

Explanation / Answer

1) at t = 0, I1 = I2 = I = V/R1 = 12/295 = 0.0407 A

UL1(0) = (1/2)*L1*I1^2

= (1/2)*203*10^-3*0.0407^2

= 1.68*10^-4 J

2) wo = 1/sqrt((L1+L2)*C)

= 1/sqrt((203 + 177)*10^-3*95*10^-6)

= 166 rad/s

3) Apply energy conservation

Qmax^2/(2*C) = (1/2)*(L1+L2)*I^2

Qmax^2 = (L1+L2)*CI^2

Qmax = sqrt((L1+L2)*C)*I

= I/wo

= 0.0407/166

= 245 micro C

4)

Q = Qmax*sin(w*t)

= 245*sin(166*3*10^-3)

= 245*sin(0.498) (use calculator in radian mode)

= 117 micro C

5)

at t2 = T/4 the energy stored in the capacitor becomes maximum

so, t2 = (2*pi/wo)/4

= (2*pi/166)/4

= 9.46 ms

6) Umax = Qmax^2/(2*C)

= (245*10^-6)^2/(2*95*10^-6)

= 3.16*10^-4 J

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