Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 002 Circuits Electronics Spring 2005 Problem Set 7 Issued 3 16 04 Due 3 30 04 Exercise 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 capacitor or inductor and what decay time results from this combination v t i t L C R1 R2 v 0 V R1 v t C R2 R1 i t R2 R1 R2 L v 0 V i 0 I i 0 I Exercise 7 2 The network shown below contains a 1 mA current source a resistor and an inductor all in parallel The network has been assembled for a long time At t 0 the current source turns off after which the common voltage v t is measured as shown below From the measured voltage determine the resistance of the resistor and the inductance of the inductor 1 0 9 0 8 Common Voltage V 0 7 v t 0 6 0 5 0 4 0 3 0 2 0 1 0 0 0 5 1 1 5 Time s 2 2 5 3 Problem 7 1 This problem examines the relation between transient responses of linear systems The network shown below is first driven by a current step at t 0 then driven by a current ramp at t 0 and finally driven by the current step plus the current ramp at t 0 In the 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 current step I t I u t Assume that v 0 0 B Find the capacitor voltage v t in response to the current ramp shown I t I tu t Again assume that v 0 0 C The step input can be constructed from the ramp input according to IStep t Show that their respective responses are related in a similar manner 1 d dt IRamp t 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 t u t This time assume that v 0 v Hint think superposition I t R C v t Problem 7 2 The circuit shown below can be used to regulate the current through an inductor Typical applications include the regulation of currents in motors solenoids and loud speakers all of which have inductive windings We will analyze the circuit assuming that it operates in a cyclic manner with switching period T During the first part of each period which lasts for a duration DT switches S1 and S4 are on while switches S2 and S3 are off During the second part of each switching period which lasts for a duration 1 D T switches S1 and S4 are off while switches S2 and 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 a cyclic steady state by t 0 at which point a new switching period begins In terms of the unknown i 0 determine i t for 0 t T B Use your result from Part A and the fact that the circuit operates in a cyclic steady state to determine i 0 Note that with this result and that from Part A i t is completely determined C Find the average value of i t over the period 0 t T Hint is it necessary to average the result from Part A or is there a faster method to find the average D Suppose that the circuit has been operating with D D1 for a time long enough to reach a cyclic steady state by t 0 Suppose that D switches to D D2 at t 0 just as a new switching period begins In this case determine i t for t 0 Hint can you use your result from Parts A and B as a particular solution over the interval 0 t S3 S1 L R V i t S2 S4 Problem 7 3 Consider the digital logic circuit shown below Model each MOSFET with the switch resistor model and let the on state resistance RON satisfy RON RPU Further assume that MOSFET M4 has a gate to source capacitance CGS Given that the inputs IN1 IN2 and IN3 cycle through the combinations 000 001 010 011 100 101 110 111 determine the average power dissipated by the logic circuit Assume that each input combination is held for the period T with T RPU CGS Make appropriate simplifications based on the inequalities for RON and T RPU RPU M4 M1 V vOUT M3 vIN1 M2 vIN3 vIN2 Problem 7 4 The network shown below contains a 1 F capacitor and a 10 mH inductor At t 0 the capacitor voltage vC is 5 V and the inductor current iL is 100 mA A Over what period do the network states oscillate B What is the maximum value that vC will reach C What is the maximum value that iL will reach D At what time after t 0 will vC first reach its maximum positive value E At what time after t 0 will iL first reach its maximum positive value iL 1 F vC 10 mH
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