Network A Network B R2 I R1 i t L R1 V i 0 io C R2 v t v 0 vo Problem 7 4 This problem examines the relation between transient responses of linear systems The network shown below is first driven by a step at t 0 then driven by a ramp at t 0 and finally driven by a stepped ramp at t 0 In the first two cases the capacitor has zero initial voltage A Find the capacitor voltage v t in response to the step shown below Assume that v 0 0 B Find the capacitor voltage v t in response to the ramp shown below Again assume that v 0 0 Hint since the step input can be constructed from the ramp input according to 1d vStep t vRamp t their respective ZSR responses are related in a similar manner dt C Finally find the capacitor voltage v t in response to the stepped ramp shown below assuming that the capacitor has the known initial voltage v 0 vo Hint think superposition V t Ramp V t Step V t Stepped Ramp Vo V 0 t 0 t t 0 R V t C v t V 0 1 t Vo 0 Part A v 0 0 Part B v 0 0 Part C v 0 vo t B Next at t T vIN turns the MOSFET off Determine both iR t and vDS t for t 0 Hint iR t is continuous at t T C Sketch and clearly label graphs of both iR t and vDS t for t 0 assuming that T 5LR RR and R X R R D The relay control circuit would be less expensive without the external resistor which may be removed from the circuit by considering the limit RX Why might such a cost reduction be unwise VS Relay LR iR Rx RR vIN vDS Problem 7 3 This problem illustrates the superposition of a zero input response ZIR and a zero state response ZSR as a means of determining the total response of a network A dx dt Solve the differential equation x S for t 0 given x 0 This is equivalent to finding the step response of a general linear first order time invariant system having a nonzero initial condition B Use the result from Part A to show that the step response of a linear time invariant first order system takes the form x t x 0 e t x 1 e t Explain why the two terms in this response are the ZIR and ZSR of the system respectively C For each network shown below find the network state at t and the network time constant note that I and V are constants Hint see Exercise 7 3 Next use the results of Part B to find the network state for t 0 You should consider whether you find the superposition of a ZIR and ZSR to be a simple and intuitive method of determining the response of a linear system Problem 7 1 At t 0 the networks shown below have zero initial state That is the capacitor voltage v t and the inductor current i t are both zero at t 0 At t 0 the voltage source produces an impulse of area and the current source produces an impulse of area Q A Derive the differential equation which relates v t to V t and i t to I t Hint consider using Thevenin or Norton equivalent networks to simplify the work B Find the capacitor voltage v t and the inductor current i t at both t 0 and t One way to find the states at t 0 is to integrate the corresponding differential equations from t 0to t 0 under the assumption that each state remains finite during that time you should justify this assumption Then substitute the initial conditions at t 0 into the results to determine the states at t 0 Try to determine the states at t through physical rather than mathematical reasoning C Next find the time constant by which each state goes from its initial value at t 0 to its final value at t Hint see Exercise 7 3 D Using the previous results and without necessarily solving the differential equations directly construct v t and i t for t 0 E Verify that the solutions to Part D are correct by substituting them into the differential equation found in Part A R1 V t R2 0 C v t R2 R1 I t V t I t Q t 0 i t L t Problem 7 2 In the circuit shown below a MOSFET and an external resistor having resistance RX are used to control the current iR in the winding of a relay Here the relay is modeled as a series inductor and resistor having inductance LR and resistance RR respectively The MOSFET may be modeled as an ideal switch A At t 0 vIN turns the MOSFET on so that vDS 0 Determine iR t for t 0 given that iR t 0 0 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6 002 Electronic Circuits Spring 2000 Homework 7 Issued 3 15 2000 Due 3 29 2000 Exercise 7 1 Consider an amplifier with an input output relation that takes the form vOUT VA vIN VB 3 where VA and VB are voltage constants Determine its output bias voltage VOUT and its small signal gain vout vin for a given input bias voltage VIN Exercise 7 2 Find the capacitance of the all capacitor network and the inductance of the allinductor network shown below C1 C2 L2 L1 C3 L3 Exercise 7 3 Each network shown below has a non zero initial state at t 0 as indicated Find the network state 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 R1 C R2 v 0 V i t L R1 C v t R2 v 0 V i t R1 R2 i 0 I R1 L i 0 I R2
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