Home / Exercises 06 / LC and some switches, take 2

LC and some switches, take 2

The questions below are due on Wednesday October 16, 2019; 11:59:00 PM.
 
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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:


Part A

Determine the energy stored in the inductor w_L(t) at t = T_1.

Develop an expression for w_L(T_1). Show your work.
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Part B

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.

Develop an expression for v(T_2). Show your work.
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Part C

Determine the energy stored in the inductor w_L(t) at t = T_4.

Develop an expression for w_L(T_4)=. Show your work.
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Part D

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 RLC switches, take 1 Part B when the latter is evaluated at t = T_5.

Develop an expressions for v(T_5)=. Show your work.
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Part E

Now let the switches move repetitively through the three-step cycle described in LC switches, take 1:

  • S1 is initially closed with S2 open,
  • next S1 is open with S2 closed
  • finally both S1 and S2 are 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 .

Develop an expression for v(t) at the end of the nth cycle. Show your work.
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