D Following Part A let vI t VIej t Also let vO t V O ej t where V O is a complex function of With these substitutions use the differential equation to find V O E Following Parts A and B find VO and from V O both as functions of F Sketch and clearly label the dependence of log VO VI and on log where is the time constant of the circuit given below Identify the low and high frequency asymptotes on the sketch VI t R1 L R2 VO t R1 R2 L R1 R2 Problem 8 3 In the network shown below the inductor and capacitor have zero states prior to t 0 At t 0 a step in voltage from 0 to VO is applied by the voltage source as shown A di at t 0 Find vC vL vR i and 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 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 O where O VO VO 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 dt Save a copy of your answers to this problem They will be useful during the pre lab exercises for Lab 3 i t vL V t L VO V t t R vR C vC Problem 8 4 The network shown below is driven in steady state by the sinusoidal input voltage vI t VIcos t The output of the network is the voltage vO t which takes the form vO t VOcos t Find VO and as functions of as follows A Using the Taylor Series expansions for ex cos x and sin x show that ejx cos x jsin x Following this recognize that cos x ejx B Show that A Bj A 2 B 2 ejarctan B A Thus the magnitude and phase of A Bj are A 2 B 2 and arctan B A respectively C Find a differential equation that can be solved for vO t Problem 8 1 The network shown below includes two switches 1 and 2 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 Switch 1 opens and it remains open until t T1 Find v t and i t for 0 t T1 B At t T1 Switch 1 closes as Switch 2 simultaneously opens They remain in these states until v t goes to zero at which time Switch 2 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 Switch 1 again opens and it remains open until t T4 Find v t and i t for T3 t T4 E Finally at t T4 Switch 1 closes as Switch 2 again simultaneously opens They remain in these states until v t again goes to zero at which time Switch 2 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 C I Switch 1 Switch 2 i t 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 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 002 Electronic Circuits Spring 2000 Homework 8 Issued 3 29 2000 Due 4 5 2000 Exercise 8 1 All networks shown below begin operation at t 0 with zero capacitor voltage or zero inductor current That is all states are zero at t 0 For each network find the network state that is the capacitor voltage or inductor current at both t 0 and t Also find the time constant by which the network state goes from its initial value at t 0 to its final value at t Finally without actually solving an appropriate differential equation find the network state for each network for 0 t Source Voltage Source Voltage R V0 R Area 0 C t 0 L t 0 Source Current Source Current I0 Area Q0 0 R R C t L t 0 Exercise 8 2 Using one 1 F capacitor and three resistors construct a two port network that has the following zero state response to a 3 V step input as shown below Provide a diagram of the network and specify the values of the three resistors vIN t 3V vIN t t Network vOUT t vOUT t 1 1 t 2ms V V 1 e 2 3 3 V 3 1 V 3 t
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