MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE 6.002 Circuits and Electronics Quiz #2 November 15, 2005 YOUR NAME___________________SOLUTIONS_________________________ Recitation Instructor / TA ________________________________________________ General Instructions: 1. Please verify that there are 15 pages in your exam booklet. 2. Please do all of your work in the spaces provided in this examination booklet. In particular, try to do your work for each question within the boundaries of the question, or on the back side of the page preceding the question. Extra pages are also available at the end of the booklet. Place the answer to each question within the appropriate answer box. 3. You may use two double-sided pages of notes and a calculator while taking this exam. For examiner’s use only: Problem Points Score Grader 1 30 2 30 3 40 Total 100Problem 1 (30 Points) For the circuits below, please find expressions for the specified voltage or current over the indicated time ranges in terms of the circuit parameters. Plot the waveform on the provided axes, and clearly identify the key parameters in your graph. (A) Consider the circuit of Fig. 1. The switch is closed for t < 0, open for 0 ≤ t < t1, and closed for t ≥ t1, where t1 = 3L/R. Find and plot the current iL(t). IDD = 1mA B = 10-3 = 1mS VGG = 2.5V VT = 1V RL = 5kΩ Figure 1 Plot iL(t) over all time, indicating important waveform parameters. iL (t), t < 0 = 0 iL (t), 0 < t < t1 =)1(21τteI−−⋅ iL (t), t > t1 = ()2612τteeI−−−(B) Consider the circuit of Figure 2, in which α > 0. The switch is open for t < 0, and closed for t ≥ 0. vC(0-) = V0. Find and plot the voltage v1(t). Figure 2 τ1 =L/2R τ2 = L/R 3R t1= t iL ()612−− eIPlot v1(t) over all time, indicating important waveform parameters. Find the Thevenin equivalent as seen by the capacitor when the switch closes: Apply VT, IT By KVL around the loop: VT = R1 (α +1) IT +R2 IT VT / IT = R1 (α +1) +R2 REQ = R1 (α +1) +R2 1) Find the time constant τ τ = REQC τ = (R1 (α +1) +R2) C 2) Find V1 at t = 0- At t =0- V1 = 0 3) Find V1 at t = 0+ At t =0+ 02)1(1)1(11 VRRRV ⋅++⋅+⋅=αα 4) Find V1 at t = ∞ At t = ∞ V1 = 0 V1(t) , t < 0 = 0 V1(t) , t > 0 CRReRRRVto⋅++⋅=⋅++⋅+⋅⋅−)2)1(1(,2)1(1)1(1αταατ V0(R1(α+1)) (R1(α+1)+R2 V1 tProblem 2 (30 Points) Searching through your 6.002 lab kit you discover a curious, purple, two terminal device that you affectionately decide to name Device Q. Taken by the moment, you set out to find the large-signal I-V response of this device. By applying voltage vQ between the two device terminals you measure the current iQ flowing through Device Q to be: iQ = A (vQ)2 for vQ > 0 where A = 1 x 10-4 A/V2 (a) What is the small signal current, iq, of Device Q if you operate it at a DC operating point VQ with a small-signal voltage excitation vq? (Please write the general expression for iq in terms of A, VQ, vq) Answer: qQqvAVi ⋅= 2 (b) You insert the Device Q into the circuit below. From previous experience, you know that the MOSFET in the circuit has a threshold voltage VT = 1V, and has an output drain-source characteristic that looks like a resistor RMOSFET = 500Ω when the gate-source voltage vGS = 15V and it is operating in its linear (triode) region. What is the value of vOUT when vIN = 0.7V?The MOSFET is off since vIN = 0.7V < VT and is = 0 => sOUTVv = vOUT = _15V_____ What is the value of vOUT when vIN = 15V? VVVARsVRAsVRAARVVRAsVRAsVRARAVRAVVRAVVRAVRAVRAVVRAVRAVVVARVVVAiRVioososososooossoosoossos52025101410152)2(2)12(2212)2(2)12(2212)12(02)()(222222=±=±+=⋅⋅⋅−+⋅⋅⋅±+=⋅⋅⋅⋅⋅−+⋅⋅⋅±⋅⋅⋅+⋅⋅⋅⋅=⋅⋅+⋅+⋅⋅⋅⋅−⋅⋅=⋅⋅+⋅⋅⋅⋅−⋅⋅=−⋅⋅=⇒−⋅== vOUT = ___5_____ (c) Ripple in the power supply causes the value of vS to fluctuate by a few percent around the intended 15V. Write the expression for vout (the fluctuation in vOUT) in terms of the fluctuation vs, when vIN = 15V. From part (a) you know that qqqvAVi⋅= 2 Ω==⇒⋅=⇒ 5005001qqqqqivrvivout = ssMOSFETqqvvRrr⋅=⋅+ 21Rq = 500Ω vs RMOSFET = 500Ω voutProblem 3 (40 Points) You are given a “LINFET” transistor whose drain current in saturation is given by: iD = B (vGS – VT) While this transistor has a linear iD - vGS relation, all other characteristics are as usual: iG = 0, and operation in saturation requires: vGS > VT and vGS – VT < vDS The transistor is used in the amplifier circuit below: (A) Assume the transistor is biased to operate in saturation. Draw the equivalent network for this circuit where the transistor is replaced by its large-signal equivalent circuit. Draw the large-signal circuit model for the amplifier RL B(VGG-VI-VT) VoutVi IDD + __ Iv(B) For the LINFET to operate in saturation X < vI < Y. Derive expressions for X and Y and give numerical answers. 1) For the MOSFET to turn on TGSVV > TGGITIGGVVvVvV−<⇒>− VvI5.1< For the MOSFET to be in saturation vGS – VT < vDS TIGGIoVVVVV −−>− TGGLDSDDVVRII −>⋅− )( TGGLTIGGLDDVVRVVVBRI−>⋅−−⋅− )( LDDTLLGGILRIVBRBRVBVR−+−+> )1()1( LLDDTLLGGIBRRIVBRBRVV−+−+>)1()1( 12.13 −−>IV 8.0>IV X = 0.8 Y = 1.5 => _________0.8 < vI < 1.5_____________ (C) Still under the saturation operation assumption, draw the small-signal equivalent network for the circuit. Assume that vI = VI + vi where VI is a constant voltage and vi is a small-signal excitation. Label all circuit elements. Make sure all controlling and controlled values are clearly shown.(D) Assuming that vI = VI + vi derive an expression for the small-signal gain vout / vi , and give a numerical answer. vout / vi =__BRL = 5____ RL Bvi vout
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