Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits & ElectronicsSpring 2005Problem Set #7Issued 3/16/04 – Due 3/30/04Exercise 7.1: Each network shown below has a non-zero initial state at t = 0, as indicated.Find the network states for t ≥ 0. Hint: what equivalent resistance is in parallel with each capacitoror inductor, and what decay time results from this combination?R1+ v(t) −CR2v(0) = VR1+v(t)−CR2v(0) = VR1i(t)LR2i(0) = IR1i(t)LR2i(0) = IExercise 7.2: The network shown below contains a 1-mA current source, a resistor and aninductor all in parallel. The network has been assembled for a long time. At t = 0 th e currentsource turns off, after which the common voltage v(t) is measured as shown below. From themeasured voltage, d etermine the resistance of the resistor and the inductance of the inductor.+v(t)−0 0.5 1 1.5 2 2.5 300.10.20.30.40.50.60.70.80.91Time [µs]Common Voltage [V]Problem 7.1: This problem examines the relation between transient responses of linearsystems. The network shown below is first driven by a current step at t = 0, then driven by acurrent ramp at t = 0, and finally driven by the current step plus the current ramp at t = 0. Inthe first two cases, the capacitor has zero initial voltage, while in the last case it does not.(A) Find the capacitor voltage v(t) in response to the curr ent step I(t) = I◦u(t). Assume thatv(0) = 0.(B) Find the capacitor voltage v(t) in response to the current ramp shown I(t) = I◦αtu(t). Againassume that v(0) = 0.(C) Th e step inp ut can be constructed from the ramp input according to IStep(t) =1αddtIRamp(t).Show that their respective responses are related in a similar manner.(D) Would the result from Part C hold if v(0) 6= 0? Why or why not?(E) Finally, find the capacitor voltage v(t) in response to the current step plus the current ramp,that is to I(t) = I◦(1 + αt)u(t). This time assume that v(0) = v◦. Hint: think superposition.I(t)+v(t)−CRProblem 7.2: The circuit shown below can be used to regulate the current through an inductor.Typical applications includ e the regulation of currents in motors, solenoids and loud speakers, allof which have inductive windings. We will analyze the circuit assuming th at it operates in a cyclicmanner with switching period T . During the first p art of each period , which lasts for a durationDT , switches S1 and S4 are on while switches S2 and S3 are off. During the second part of eachswitching period, which lasts for a duration (1 − D)T , switches S1 and S4 are off while switches S2and S3 are on. Note that 0 ≤ D ≤ 1.(A) Assume that D is constant and that the circuit has been operating long enough to reach acyclic steady state by t = 0, at which point a new switching period begins. In terms of theunknown i(0), determine i(t) for 0 ≤ t ≤ T .(B) Use your result from Part (A), and the fact that th e circuit operates in a cyclic steady stateto determine i(0). Note that with this result, and that from Part (A), i(t) is completelydetermined.(C) Find the average value of i(t) over the period 0 ≤ t ≤ T . Hint: is it necessary to average theresult from Part A, or is there a faster method to find the average?(D) Suppose that the circuit has been operating with D ≡ D1for a time long enough to reach acyclic s teady state by t = 0. Suppose that D switches to D = D2at t = 0, just as a newswitching period begins. In this case, determine i(t) for t ≥ 0. Hint: can you use your resultfrom Parts (A) and (B) as a particular solution over the interval 0 ≤ t?VS3S4S2S1Ri(t)LProblem 7.3: Consider the digital logic circuit shown below. Model each MOSFET with th eswitch-resistor model, and let the on-state resistance RONsatisfy RON≪ RPU. Further assumethat MOSFET M4 has a gate-to-source capacitance CGS. Given that the inputs IN1, IN2 and IN3cycle through the combinations 000, 001, 010, 011, 100, 101, 110, 111, determine the average powerdissipated by the logic circuit. Assume that each in put comb ination is held for the period T withT ≫ RPUCGS. Make appropriate simplifications based on the inequalities for RONand T .VRPUM4+vOUT−M2M1RPUM3vIN1vIN2vIN3Problem 7.4: The network shown below contains a 1-µF capacitor and a 10-mH inductor. Att = 0, the capacitor voltage vCis 5 V, and the inductor current iLis 100 m A.(A) Over what period do the network states oscillate?(B) What is the maximum value that vCwill reach?(C) What is the maximum value that iLwill reach?(D) At what time after t = 0 will vCfirst reach its maximum positive value?(E) At what time after t = 0 will iLfirst reach its maximum positive value?1 µFiL10
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