Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits & ElectronicsSpring 2004Final Exam18 May 2004Name:Instructor: Kassakian Kassakian Wilson Berggren BerggrenTime: 10 11 12 1 2• Please put your name in the space provided a bove, and circle the name of your recitationinstructor together with the time of your recitation.• Do your work for each question within the boundaries of the question. When finished, writeyour answer to each question in the corresponding answer box that follows the question.• This is a closed-book exam, but calculators and a single two-sided page of notes are allowed.• You may review and take back your final exam on or after Thursday May 20. However,overall grades for 6.002 will not be determined until Friday May 21, a nd will be most easilyobtained through WEBSIS.• Good luck!Problem 1 Problem 2 Problem 3 Problem 4 Total GradeProblem 1 – 25%This problem consists of fourteen independent parts, having a total of twenty five sub-parts. Allop-amps in this problem are ideal. Note that all sub-parts are weighted equally at only 1% each. So,if you get stuck on one sub-part it might be wise to skip that sub-part and move on until later.(1A) Determine the Thevenin equivalent voltag e vTHand resistance RTHof the network shownbelow.vTH: RTH:4A12Ω7Ω2Ω2Ω+-(1B) Determine the ratio vO/vS.vO/vS:+−3Ω2Ω6Ω1Ωvsvo-+(1C) Determine vC(0+), the time constant τ that determines the response of the network, and anexpression for vC(t) for t ≥ 0. Note that u0(t) denotes the unit impulse function.vC(0+): τ:vC(t ≥ 0):R2R1Qu0(t)C+-vcvc(0-)=V0(1D) Determine iL(∞), and the time constant τ that determines the response of the network.iL(∞): τ:+−2Ω 4Ω2Ω3Ω10hiLV0u-1(t)(1E) Determine the Thevenin equivalent voltag e vTHand resistance RTHof the network shownbelow.vTH: RTH+−1KΩ180Ω10Vib49ib+-(1F) Determine the maximum value of iL.max(iL):10µf10µh+-vciLcloses@t=0vc(0)=10V iL(0)=0(1G) For the logic circuit shown below, determine a Boolean expression for the logical output Zas a function of the logical inputs A, B and C.Boolean Expression For Z:+−+−+−vAvCvBRvzVs+-(1H) Determine iNL.iNL:+_vIN=3VSeeGraph+_vNLiNL1ΚΩvNLiNL2mA4V10-3(1I) Determine vO.vO:−+voαvgs3+-vA+-vgsRVs+-(1J) Determine vO(t) in the sinusoidal steady state.vO(t):+-10cos(ωt)LC−+vo+-(1K) Determine dvC/dt and diL/dt at t = 0+, that is, just after the switch opens.dvCdt(0+):diLdt(0+):1µf10mh+-vciLopens@t=0vc(0-)=0ViL(0-)=1A2Ω(1L) Determine v1, v2and i1.v1: v2:i1:v1i110-3-10v2i2-10-351KΩ+_+_v1v2i1i2(1M) The voltage vCin the network below reaches a steady state in a finite time. Determine itssteady state value and the time t at which it reaches that value.Steady-state vC: t:Networkt > 0i+_v1µF+_vCvC(0) = 0-10-32x10-310Vvi0.02A0.01A5V(1N) Determine the Thevenin equivalent volta ge vTHand complex impedance ZTHof the networkshown below, which operates in the sinusoidal steady state. The Thevenin equivalent voltageis defined by its amplitude VTHand its phase φTHsuch that vTH= VTHcos(ωt + φTH).VTH: φTH:ZTH:Icos(ω t)LRC+_Problem 2 – 25%The circuit shown below is a free-running oscillator that provides a square wave output at v2. Allop-amps are ideal except that they can saturate at the power supply voltages ±VS. Note that, forsimplicity, the power supply connections to the op-amps are not shown in the figure.+_#1+_#3+_#22R1R1R1CR2R3++__v1v2(2A) Consider Op-amps #1 and #2. Assume that they do not saturate, and determine v1(t) as afunction of v2(t), R1and C.v1(t):(2B) Consider Op-amp #3, which is always saturated. Determine the value of v1at which v2switches from +VSto −VS, and the value of v1at which v2switches from −VSto +VS.v1(v2↓): v1(v2↑):(2C) Determine the period over which the oscillator oscillates.Period:Problem 3 – 25%This problem concerns the op-amp amplifier shown in Figure A b elow. The op-amp in that amplifieris modeled by the network shown in Figure B below.+_vINvOUTRR+_+_(A) Amplifier+_A1v1R1C1+_v2R2C2+_v3+_ A2v2+_A3v3+_v4+_v1v+v--InputInputOutput(B) Model(3A) Assume that vINhas been constant for a very long time. Determine vOUTin terms of vIN, Rand the parameters of the op-a mp model.vOUT:(3B) Consider the op-amp model in Figure B. Let A1= 1, A2= 2, A3= 5 and R1C1= R2C2=0.01 s. Further, assume that v1takes the form v1(t) = ℜ(ˆV1ejωt) so that v4takes the formv4(t) = ℜ(ˆV4ejωt), whereˆV1andˆV4are both complex constants. In t his case, determine theratioˆV4/ˆV1.ˆV4/ˆV1:(3C) Consider the op-amp amplifier in Figure A. Assume that vINtakes the form vIN(t) =ℜ(ˆVINejωt) so that vOUTtakes the form vOUT(t) = ℜ(ˆVOUTejωt), whereˆVINandˆVOUTareboth complex constants. In this case, determine the ratioˆVOUT/ˆVINfor the model parame-ters given in Part B.ˆVOUT/ˆVIN:Problem 4 – 25%This problem analyzes a multi-stage MOSFET amplifier. To simplify the analysis, assume that allMOSFETs are linear, have a threshold voltage of 0 V, and operate in their saturation region. Thus,all MOSFETs, both n-channel and p-channel, operate according to iD= KvGS, where iDis definedto be positive into the drain.(4A) The circuit shown below functions as a current source for its load. Determine the sourcecurrent ISas a function of R1, R2, VSand K.IS:+_+_LoadISR1R2VSVSGDS(4B) Determine the voltage v3in the circuit shown below in terms of v1, v2, R3, VS, ISand K.v3:V3+_+_+_+_VSVSV1V2ISR3+_DSG GDS(4C) Determine the voltage v4in the circuit shown below in terms of v3, R4, VS, and K. Notethat the MOSFET in the circuit is a p-channel MOSFET so that vGS< 0, vDS< 0 andiD= KvGS.v4:+_+_VSVS+_V3R4DGS+_V3(4D) The amplifier shown below is a combination of the circuits analyzed in Parts A, B and C ofthis problem. For this amplifier, determine vOUTas a function of v1, v2, R1, R2, R3, R4, VSand
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