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MIT 6 002 - Study Guide

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Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits & ElectronicsSpring 2005Problem Set #8Issued 3/30/05 – Due 4/6/05Exercise 8.1: Using a 1-nF capacitor and two resistors, construct a two-port network thathas the response shown below to a 1-V step input; assume that the capacitor voltage is zero priorto the step. Provid e a diagram of the network, and specify th e values of the two resistors.1 V u(t)+(1 V − 0.5 V e−t/(1 µs))u(t)−Exercise 8.2: Repeat Exercise 8.1 given that the allowable components are now a 1-mHinductor and two resistors; assume that the inductor cu rrent is zero prior to the step.Problem 8.1: The network shown below includes two switches: S1 and S2. Pr ior to t = 0,both s witches are closed, and the capacitor voltage v(t) and inductor current i(t) are both zero.(A) At t = 0, S1 opens, and it remains open until t = T1. Find v(t) and i(t) for 0 ≤ t ≤ T1.(B) At t = T1, S1 closes as S2 simultaneously opens. They remain in these states until v(t) goes tozero, at which time S2 closes. Define the time at which v(t) goes to zero as t = T2. DetermineT2, and find v(t) and i(t) for T1≤ t ≤ T2.(C) Both switches remain closed until t = T3. Find v(t) and i(t) for T2≤ t ≤ T3.(D) At t = T3, S1 again opens, an d it remains open until t = T4. Find v(t) and i(t) for T3≤ t ≤ T4.(E) Finally, at t = T4, S1 closes as S 2 again simu ltaneously opens. They remain in these statesuntil v(t) again goes to zero, at w hich time S2 closes. Define the time at which v(t) again goesto zero as T5. Determine T5, and find v(t) and i(t) for T4≤ t ≤ T5.(F) Sketch and clearly label v(t) and i(t) for 0 ≤ t ≤ T5.I+ v(t) −Ci(t)LS1 S2Problem 8.2: This problem is a continuation of Problem 8.1. It explores the use of energyconservation to analyze the operation of the network described therein.(A) Determine the energy stored in the capacitor at t = T1.(B) The energy stored in the capacitor at t = T1is transferred to the inductor at t = T2. Use thisfact to determine i(T2). This answer should match the answer to Part B of Problem 8.1.(C) Determine the energy stored in the capacitor at t = T4.(D) Use energy conservation to determine the energy stored in the inductor at t = T5, and thendetermine i(T5). This answer should match the answer to Part E of P roblem 8.1.(E) Now let the switches move repetitively through the cycle in which they both begin closed,then S1 opens, then both switches simultaneously change state, then S2 closes to return to thebeginning of the cycle. Assume that in each cycle S1 is open alone for the duration T . Further,assume th at S2 always closes when v(t) reaches zero. Assuming that v and i are initially zero,determine i at the end of the nth switching cycle in terms of n, C, L, T and I.Problem 8.3: In the network shown below, the ind uctor and capacitor have zero current andvoltage, respectively, prior to t = 0. At t = 0, a step in voltage from 0 to V◦is applied by th evoltage source as shown.(A) Find vC, vL, vR, i anddidtjust after the step at t = 0.(B) Argue that i = 0 at t = ∞ so that i(t) has no constant component.(C) Find a s econd-order differential equation which describes the behavior of i(t) f or t ≥ 0.(D) Following (B) the current i(t) takes the form i(t) = I sin(ωt + φ)e−αtfor t ≥ 0. Find I, ω, φand α. Hint: first find ω and α from the differential equation, and then find I and φ from theinitial conditions; alternatively, solve this problem by any method you wish.(E) Suppose that the input is a voltage impulse with area Λ◦where Λ◦= τ V◦, V◦is the amplitudeof th e voltage step sh own below, and τ is a given time constant. Find the response of thenetwork shown below to the impulse. Hint: before solving this problem directly, consider therelation between step and impulse responses.Save a copy of your answers to this problem. They will be useful during the pre-lab exercises forLab #3.V◦u(t)+ vL(t) −Li(t)+vR(t)−R− vC(t)


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MIT 6 002 - Study Guide

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