Question

d2iL(t)dt2+1LCiL(t)=ALCδ(t−9.0)
a-What is the natural frequency, in Hertz, of this circuit?
0.249
b-For the remaining parts of the question, you should round the above value off to two decimal places to make calculations easier.


c-At the initial time what is the total energy, in Joules, stored in the circuit?


d-At the time just before the impulse happens t=9.0s− what is the total energy, in Joules, stored in the circuit?


e-At the time just before the impulse happens what is the current iL(9.0s−) , in Amperes, through the inductor?


f-At the time just before the impulse happens what is the voltage vC(9.0s−) , in Volts, across the capacitor?


g-At the time just after the impulse happens what is the current iL(9.0s+) , in Amperes, through the inductor?


h-At the time just after the impulse happens what is the voltage vC(9.0s+) , in Volts, across the capacitor?


i-At the time just after the impulse happens what is the total energy, in Joules, stored in the circuit?

Answer 1

There is something not right about what you have posted, maybe screen grab the actual text?

The underlying type of DE, here a non-hom DE for an L-C circuit, is:

- \(\qquad x'' + \omega^2 x = \gamma ~ \delta (t - 9)\)

The solution is:

- \(\qquad x = c_1 \sin(\omega t) + c_2 \cos( \omega t) + \dfrac{\gamma \mathcal{H}(t - 9) sin(\omega (t - 9))}{\omega}\)

...with \(\mathcal{H}\) being the Heaviside/ unit step function

Solution uses a Laplace T/f.

So you still have that \(\dfrac{1}{\sqrt{ L C }}\) resonance thing going on