Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 002 Circuits Electronics Spring 2005 Problem Set 8 Issued 3 30 05 Due 4 6 05 Exercise 8 1 Using a 1 nF capacitor and two resistors construct a two port network that has the response shown below to a 1 V step input assume that the capacitor voltage is zero prior to the step Provide a diagram of the network and specify the values of the two resistors 1 V u t 1 V 0 5 V e t 1 s u t Exercise 8 2 Repeat Exercise 8 1 given that the allowable components are now a 1 mH inductor and two resistors assume that the inductor current is zero prior to the step Problem 8 1 The network shown below includes two switches S1 and S2 Prior to t 0 both switches 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 to zero at which time S2 closes Define the time at which v t goes to zero as t T2 Determine T2 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 and it remains open until t T4 Find v t and i t for T3 t T4 E Finally at t T4 S1 closes as S2 again simultaneously opens They remain in these states until v t again goes to zero at which time S2 closes Define the time at which v t again goes to 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 v t i t C I S1 S2 L Problem 8 2 This problem is a continuation of Problem 8 1 It explores the use of energy conservation 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 T1 is transferred to the inductor at t T2 Use this fact 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 then determine i T5 This answer should match the answer to Part E of Problem 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 the beginning of the cycle Assume that in each cycle S1 is open alone for the duration T Further assume that 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 inductor and capacitor have zero current and voltage respectively prior to t 0 At t 0 a step in voltage from 0 to V is applied by the voltage source as shown A Find vC vL vR i and di dt just after the step at t 0 B Argue that i 0 at t so that i t has no constant component C Find a second order differential equation which describes the behavior of i t for t 0 D Following B the current i t takes the form i t I sin t e t for t 0 Find I and Hint first find and from the differential equation and then find I and from the initial conditions alternatively solve this problem by any method you wish E Suppose that the input is a voltage impulse with area where V V is the amplitude of the voltage step shown below and is a given time constant Find the response of the network shown below to the impulse Hint before solving this problem directly consider the relation between step and impulse responses Save a copy of your answers to this problem They will be useful during the pre lab exercises for Lab 3 vL t i t L V u t R C vC t vR t
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