LC and some switches, take 2
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For the questions below, use pi for the constant \pi. You can use the standard trigonometric functions sin(arg), cos(arg), tan(arg), and inverse functions asin(arg), acos(arg), and atan(arg). You can use sqrt(arg) or do the root explicitly in python arg**0.5.
This problem is a continuation of LC switches, take 1. It explores the use of energy conservation to analyze the operation of the network described therein, repeated here below:
Note that it's entirely possible to get the answers to this problem almost directly from RLC switches, take 1. If you do that, though, you don't learn anything...
Determine the energy stored in the inductor w_L(t) at t = T_1. Enter T_1 as T_1.
The energy stored in the inductor at t = T_1 is fully transferred to the capacitor at t = T_2. Use this fact to determine v(T_2). Note that this answer should match your answer to LC switches, take 1 Part B when the latter is evaluated at t = T_2.
Determine the energy stored in the inductor w_L(t) at t = T_4.
Use energy conservation to determine the energy stored in the capacitor at t = T_5, and then determine v(T_5). Note that this answer should match your answer to LC switches, take 1 Part B when the latter is evaluated at t = T_5.
Now let the switches move repetitively through the three-step cycle described in LC switches, take 1: S1 initially closed with S2 open, next S1 open with S2 closed, finally both S1 and S2 open. Assume that in each cycle S1 remains closed for the duration T. Further, assume that S2 always opens when i(t) reaches zero. Assuming v(t) and i(t) are initially zero, determine v(t) at the end of the nth switching cycle in terms of n, C, L, T and V .