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MIT 6 002 - Study Notes

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Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits and ElectronicsSpring 2003Handout S03-037 - Homework #7Issued: Wed. Mar 19Due: Fri. Apr 4Problem 7.1: The circuits below are driven by either step functions or impulse functions. Ineach case determine the initial (t = 0+) and final (asymptotic) values of the designated voltagesand/or currents. Label your answers clearly.(A)+−Vu-1(t)LRiv+-(B)Iu-1(t)LRiv+-(C)Qu0(t)CRiv+-(D)λu0(t)C2Rv+-+−2R(E)Qu0(t)LRiv+-RProblem 7.2: Pick any three of the five circuits shown in Problem 7.1. For each of your choices,sketch and dimension the indicated voltages and currents for t > 0. Evaluate time constants interms of circuit elements. Label your drawings clearly, including the designation (A), (B) · · · (E)of your choices.Endeavor to do this problem without formally solving the differential equations.Problem 7.3: The gray box shown below contains only linear circuit elements and satisfiesthe strict definition of linearity.+−GRAYBOXbb’+-vO+-vIu0(t)aa’When the box is initially without stored energy and is driven by a unit voltage inpulse at theterminals aa0as shown, the response of the voltage vOfor t > 0 isvO(t) =23e−tτfor t > 00vO(t)2/3τt(A) Determine the response vO(t) when the input vIat aa0is a step of amplitude V .vI= V u−1(t)(B) The input to the box is shown below.0vI(t)VTtDetermine the output voltage vO(t) for t > 0.Note that a response to a delayed input can be written asv(t) = u−1(t − T )f(t − T )where f(t) is the response to an excitation at t = 0 and T is the time the input is delayed.The multiplier u−1(t − T ) ensures that there is no reponse for t < T .Hint: Resolve the input into the sum of three inputs, each of which is a scaled singularityfunction.Problem 7.4: The LC circuit below is driven by an impulse:LCiv+-Qu0(t)(A) Determin v(0+) and i(0+).(B) At t = 0+: What is the sign of the first derivative of v?What is the sign of the first derivative of i?(C) Note that for t > 0 the circuit is:LCiv+-Write a differential equation for v(t) or i(t) and solve it. Express both v(t) and i(t) as functionsof


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MIT 6 002 - Study Notes

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