Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer Science6.002 – Circuits and ElectronicsSpring 2003Handout S03-065 - Final ExamTuesday May 20, 2003Name:Recitation Instructor (circle one):Baldo Hutchinson Kolodziejski Schindall WilsonRecitation Hour (circle one):9 10 11 12 1 2NOTE THAT PROBLEMS HAVE WEIGHTS RANGING FROM 10 TO 16Problem Points Score Grader1 162 103 124 125 106 127 148 14Total 1001Name:PROBLEM 1Joe Nerdle, who took 6.002 some time ago, wants to assemble a three-input AND gate. He findssufficient identical MOSFETS of unknown specifications.The several parts below are independent.(A) In fifty words or less, describe measurements which Joe can make to determine the thresholdvoltage VTand the on resistance RONof the FETs. Include a circuit diagram if you wish.Joe establishes a static discipline in which a logical one corresponds to v > 4 volts and a logicalzero corresponds to v < 1 volt. A three input NAND gate and the switch-resistor model for theFET are shown below:7kΩvDvCvBvA+5VRONvGSGDS+-SWThe switch SW isopen for vGS<VTclosed for vGS>VT(B) Write a Boolean expression which describes the logical function of this gate.2Name:For the remaining parts of this problem, use these values:RON= 1kΩ, VT= 1V(C) When vA= vB= vC= 5V, determine vD.(D) Because Joe wants an AND gate, he connects the output of the gate above to a FET inverteras shown below:7kΩvDvCvBvA+5VvGS+-7kΩ+-When he builds this circuit hefinds that vDis low for all in-put conditions. Explain.(E) In an effort to understand the reason the AND gate does not work, Joe isolates the inverterand connects an oscilloscope to its output as shown below.vO+5V7kΩ+-verticalinputvO(t)+5V7kΩ+-CC=2x10-11FHe can’t find a scope probe so he uses a long piece of coaxial cable. This unfortunate arrange-ment provides a capacitive load of C = 20pF = 2 × 10−11F to the inverter. The resultingcircuit is the one shown above right.Sketch and lab el the waveform vO(t) which Joe sees on the scope when the switch changes theinput voltage from +5 V to 0 V at t = 0. Also sketch and label the waveform vO(t) whichJoe sees on the scope much later (t > 5τ) when the switch changes the input voltage from 0 Vto +5 V at t′= 0. Show asymptotes and determine the time constant. Use the coordinatesprovided on page 4.3Name:vO(volts)0+5-5vO(volts)0+5-5Waveform when switch moves from 0 volts to +5 volts at t’=0Waveform when switch moves from +5 volts to 0 volts at t=0t(seconds)t’(seconds)4Name:PROBLEM 2The circuit below contains both dependent and independent sources.+−R1R2R22v+-va a’IDetermine a Thevenin equivalent circuit at terminals a-a′. Specify polarities and element values.5Name:PROBLEM 3A FET used as an amplifier is shown in the circuit below:R1R2RLC2C1vivO=VO+vo+15V+-+-The capacitor C1couples theincremental input voltage vito the gate. Because it is anopen circuit for DC, vidoesnot affect VGS, the operating-point value of the gate-to-source voltage.R1+ R2= 500kΩ The FET is described by:iD=0 for vGS< VTK2(vGS− VT)2for vGS> VTandvDS> (vGS− VT)K = 0.4mAV2VT= 2V(A) Determine VGSand RLsuch that the operating point of the amplifier is:VO= 7V ID= 0.2mA(B) Determine the incremental transconductance gmof the FET at the operating point defined inpart (A).6Name:(C) Given that R1+ R2= 500kΩ, determine R1and R2such that the desired value of VGSisestablished.(D) Employing the usual incremental model for the FET, sketch and label a complete incrementalmodel for the circuit. Label elements of the model by their symbols, NOT by their numericalvalues.7Name:PROBLEM 4A quartz piezoelectri c crystal exhibits a strong (high Q) resonance, which is why it is used as timeor frequency standards in clocks, tuners, etc.circuit symbol forRSLCequivalent circuit fora quartz crystala quartz crystal(A) A quartz crystal is used in the filter below. Determine the transfer function of this filter. Thatis, findA(s) =VOVIwhere s = jω. Express your result as a ratio of polynomials in s.RSLC+-+-VIVOLet R=RSR8Name:For a specific set of element values in this circuit, the transfer function is:A(s) = 2.5 ×103ss2+ 5 × 103s + 0.25 × 1014(for s in sec−1)(B) Sketch the frequency dependence of the magnitude and phase ofA(jω) on the coordinatesprovided on page 10. Determine and label the resonant frequency ω0, the magnitude A0andphase ofA(jω0) at that frequency, and the high and low frequency asymptotes of magnitudeand phase. Determine the bandwidth of the resonance defined by the frequencies where |A| =A0/√2.9Name:∠A (radians)ω (radians)106107108109105104|A| π/4π/2-π/20-π/410.10.010.0010.00010.0000110Name:PROBLEM 5The circuit below is driven by vI(t) = 20 cos(ωt):vI(t) vO(t)R1R2C+-+-R1=R2=1kΩ(=103Ω)C=0.05µF(=5x10-8F)Using the indicated element values, the transfer function is:VoVi= 0.5jω + 2 × 104jω + 104Determine vO(t) for ω = 104sec−1.11Name:PROBLEM 6The circuits below function as filters in the sinusoidal steady state.+-+-ViVo+-+-ViVo+-+-ViVo+-+-ViVo+-+-ViVo+-+-ViVo+-+-ViVo+-+-ViVoLabel those which are Low-Pass Filters withLPLabel those which are High-Pass Filters withHPLabel those which are Band-Pass Filters withBPLabel those which are Band-Stop Filters withBSExtra space is provided on page 13.Grading will be based on (# correct×3−# incorrect)12Name:This page intentionally left blank.13Name:PROBLEM 7The op-amp circuit below employs a diode, which may be assumed ideal, and an n-channel FET.The op-amp has very high gain. The characteristics of both are shown.R2−++10V-10VR1+-vI+10V+-vOR1=6.2kΩR2=2.7kΩFET:iDvDSvGS+--+iD=0 vGS<0(K/2)vGS2vGS>0 vDS>vGS VT=0K=0.1mA/V2Diode:+-vi0ivDGSDGSSketch the nonlinear transfer characteristic vOvs. vIon the coordinates below for−20 V < vI< 20 V.010 20-10-2010-10vI(volts)vO(volts)Hint: Start from vI= 0 and let vIbe of either polarity.14Name:PROBLEM 8The op e rational amplifier shown below has very high gain. The supply voltages (at which theamplifier saturates) are ±10 volts. The circuit is an oscillator. Assume that the switch is open forall t < 0.−+SwitchvORRv-v+RAll voltages are definedwith respect to ground.R=10kΩ(=104Ω)L=10mH(=10-2H)LAt t = 0 the switch closes and stays closed.(A) Determine and sketch on the axes on the next page the waveforms of v+, v−and vOfor twofull cycles of the oscillation.15Name:v+(volts)010-10t8642-8-6-4-2v-(volts)010-10t8642-8-6-4-2vO(volts)010-10t8642-8-6-4-2(B) What
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