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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.002 { Electronic Circuits
Fall 2000
Final Exam
Please write your name in the space provided b elow, and circle the name of your recitation instructor and the time of your recitation.
Please verify that there are 19 pages in your exam.
To the extent p ossible, do all of your work on the pages contained within this exam. In particular, try to do your work for each question within the b oundaries of the question, or on the back side of the page preceding the question.
You may use three double-sided pages of notes while taking this exam.
Final grades in 6.002 will not b e given out by phone or by e-mail. Rather, they should b e available through WEBSIS by Decemb er 22. You may review and take back your nal exam at the b eginning of IAP in January 2001 from Ro om NE43-624.
Go o d luck!
Problem Score 1 2 3 4 5 6 7
Total
Name:
Instructor: Senturia Wilson Parker Hagelstein Sussman Time: 9 10 10 11 11 12 12 2 3
Problem 1 { 15 Points
The Op-Amp circuit shown b elow is very similar to the standard non-inverting Op Amp except that RL is some external resistor, and we are interested in showing that the current through RL is nearly constant, regardless of the value of RL , that is, the circuit acts like a current source for driving RL.
R (^) L
R 2
A i (^) L
A’
-
+
v (^) I +-
(A) Using the Op-Amp mo del shown b elow, derive an expression for iL in terms of vI , A, R 2 and RL. Show that this expression for iL b ecomes indep endent of RL as A approaches in nity.
v (^) O
v +
v
v
v
v (^) O
Problem 2 { 15 Points
This problem involves the circuit shown b elow. You are given that R = 1 , C = 1 F , and
K = (^12)
Amps Volt^2
2 R
R
2 R
v C
i D
v D
V A + v a
v a is “small”
Characteristics of nonlinear device: when
when
i D = 0 v D ≤ 0
i D = K v D^2 v D > 0
C
i D
v D
(A) Find the op erating p oint voltage VD and the op erating p oint current ID in the circuit shown ab ove. Assume for this part that VA = 12 V.
(B) Now assume that VA is changed such that a new op erating p oint ID 0 and VD 0 is obtained, where VD 0 = 1 V. (Note that this op erating p oint may di er from your answer in Part A.)
Draw the small-signal circuit, substituting numeric values where p ossible. Lab el the small-signal capacitor voltage vc and the small-signal device voltage vd. (Recall, the small signal mo del for a capacitor is the capacitor itself.)
Problem 3 { 15 Points
A nonlinear device with voltage vd and current id has a characteristic given by
id = I e vd
where vd > 0 and id > 0.
+^ v^ d
i (^) d
For all parts of this problem assume that the op erational ampli ers are ideal, that they are op erating in the active region and that the nonlinear device is op erating with p ositive values of vd and id.
(A) For the nonlinear element connected as shown b elow, nd an expression relating vout to vin.
v in
i d + v^ d
v out
R
(B) For the nonlinear element connected as shown b elow, nd an expression relating vout to vin.
v (^) in
v (^) d + i^ d v (^) out
R
(C) For the op erational ampli er circuit shown b elow, nd an expression relating vout to vin 1 , vin 2 , and vin 3.
v (^) in 1 v (^) out v (^) in 2
v (^) in 3
R
R
R
R
Problem 4 { 15 Points
Anna Logue, a UROP student working for Professor S., has just found a rather strange piece of circuitry in the trash just outside the oce of Prof. A. The circuit has three exp osed terminals and it has exactly one resistor, one inductor, and one capacitor, but she cannot see how they are connected, b ecause the connections are protected by ep oxy. She can read the capacitor lab el: it says that the capacitor has a capacitance of 1 F (Observation O1). However, the other lab els are not readable.
Being a naturally curious MIT student, Anna takes the circuit to the 6.002 lab to see what can b e determined ab out this strange device by exp eriment. Anna lab els the terminals with the letters \x", \y ", and \z " with a marker p en. She then takes several measurements. First, Anna measures the resistances b etween the terminals of the mystery circuit with an Ohmmeter, and makes the following observations:
x y : in nity (Observation O2)
y z : in nity (Observation O3)
z x: 40 (Observation O4)
Next, Anna uses a signal generator, which may b e mo deled as a voltage source, to apply a 1 Volt, 100 Hz, square-wave signal from the x terminal to the y terminal. She uses her oscilloscop e to display the voltage from z to y sup erimp osed on the square wave, at the same scale as the square wave, as shown in the gure b elow (Observation O5). As you can see, she observes that the waveform from z to y follows the square wave, but there is ringing with a cycle time of ab out 1 ms.
0
0 0.005 0.01 0.015 0.
(A) Circle the circuit b elow which b est re ects the contents of the mysterious circuit. Assume that the inductor and capacitor are ideal. Justify your choice with a two or three sentence explanation.
x
y
z x
y
z
x
y
z x
y
x z
y
z
x
y
z
(B) What is the inductance (approximately) of the inductor that is in the circuit? What is the resistance of the resistor?
Problem 5 { 10 Points
For the network shown b elow, determine an expression for the indicated variables for t > 0 and provide clearly lab eled sketches of your result. Assume that the Op-Amps are ideal and that they are op erating in the active region. Further assume that the initial current through the inductor is zero, and that the initial voltage across the capacitor is also zero.
= t^ >^0
10K
vi +
10mH
5K
10K
0.001 μ F
v 1 1 2 3
vi 2
t ( μ sec)
v 1
t ( μ sec)
v 0
v 1 v 0 = t^ >^0
v^ + 0
t ( μ sec) 0 0
Problem 6 { 15 Points
Lo oking to recover from a disastrous pro duct intro duction, the marketing team of Yikes Inc. has changed the company name to Yehaa Inc. Furthermore, they hire several MIT undergraduates with a 6.002 background in the hop es they will turn the company around. One of the engineers discovers a new three-terminal device which can b e manufactured at a lower cost than the MOSFET. This device has the characteristics shown b elow:
v DS (volts)
i DS (mA) v CS ≥ V T
v CS < V T
C
S
D
i C = 0
v DS
V T = 3V
i DS
v CS
(0V, 0mA)
(1V, 1 )
---mA
(1V, 1 )
------mA
As depicted in the gure, when vC S VT , the vD S versus iD S relationship is a straight line passing through the p oints (0V ; 0 mA) and (1V ; 18 mA).
When vC S < VT , the vD S versus iD S relationship is a straight line passing through the p oints (0V ; 0 mA) and (1V ; 481 mA).
The current into no de C of the device is always zero. In a rare display of insight, the marketing group names this device the LOSFET.
Based on their exp erience with MOSFETs, the Yehaa Inc. engineers use the LOSFET to build the inverter circuit shown b elow.
v (^) I C
V S =5V
v (^) O
12K
D
S
(C) In a given situation, a Yehaa inverter drives a capacitive load with capacitance C = 1 pF as shown in the gure b elow. Assume that the input to the inverter is at 5V and that the output is at some voltage VLO W. Supp ose that the input switches instantaneously to 0V. Find the time taken by the output of the inverter to rise to VO H following the 5V to 0V transition at the input. Express your answer in terms of VLO W and VO H.
C
v (^) I C
V S =5V
v (^) O
12K
D
S
Problem 7 { 15 Points
Consider the ampli er circuit shown b elow. Assume that the MOSFET op erates in saturation so that
iD S =
K
(vGS VT )^2
v GS
V S
R L
V S
R 1
R 2
G v DS
D
+ S
(A) A small-signal mo del for the MOSFET in the ampli er circuit is shown b elow. Given that VS = 10 V , VT = 1 V , and K = 1 mA=V 2 , determine a constraint on the values of R 1 and R 2 such that gm = 4 mA=V.
D
v (^) gs
G
i (^) ds = g (^) m v (^) gs
S
(C) Now assume that the ampli er from Part B is driven by a small-amplitude input of the form vi = Vi ej^!^ t^. Further assume that RL = RO = R
R 1 = R 2 = 2 R
CI = CO = C
Determine the complex small-signal gain of the ampli er H (j! ) = V Voi in terms of C ; R ; gm 0 ; and!. What is the magnitude of the small-signal gain of the ampli er as! approaches in nity.
