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Midterm 1
Midterm 1 – Tuesday Oct. 12, 2004, 12:40-2:00. Last
names beginning with A-L in F295 Haas; M-Z in Sibley
Auditorium, Bechtel Center.
Covers through Ch. 3, topics of HW4, concepts of first
3 labs. GSIs will review material in discussion sections
this week. Bring photo-ID, no books, no cell phones, you
may bring one 8-1/2 by 11 sheet of notes, you may bring a
calculator, and you don’t need a blue book.
EECS40, Fall 2004
Lecture 11, Slide 1
Prof. White
Homework #5
Assigned Monday 4 Oct. 2004; due by 5 pm Thursday 7 Oct. in the EE40 box in room
240 Cory. No late homework accepted. Be sure your name is legible (please print
it) and make your steps clear to the reader. Note that graded homeworks that are
not picked up in discussion sections are put in the vertical bins in front of you as
you enter room 497 Cory.
Problem 1. Verify (i.e., derive) the expression for the magnitude of the transfer function
for a high-pass first-order RC filter (one resistor, one capacitor). Notes: (1) This is
also a question in the Pre-Lab for the RC Filter and LabVIEW lab that runs from
Tuesday 5 Oct. through Monday 11 Oct. (2) The high-pass filter transfer function
expression is given in the lab write-up. (3) Also note that the corrected notes for
Lecture 10 are available on the web.
Hambley 3rd ed. : Prob. 2: P4.10; Prob. 3: P4.14; Prob. 4: 4.24.
EECS40, Fall 2004
Lecture 11, Slide 2
Prof. White
EECS 40 Readings
Readings for Midterm 1:
Hambley 3rd Ed.
Ch. 1 -- entire
Ch. 2 -- entire
Ch. 3 – all except Secs. 3.6 and 3.7
Also review concepts from Labs 1, 2 and 3
EECS40, Fall 2004
Lecture 11, Slide 3
Prof. White
Reading for the future (partial list):
(See “Readings” list on web site)
EECS40, Fall 2004
Lecture 11, Slide 4
Prof. White
EE40: Running Checklist of Electronics Terms
5.10.04 – Dick White
Terms are listed roughly in order of their introduction. Most definitions can be found in
your text.
TERM
Charge, current, voltage, resistance , conductance, energy, power
Coulomb, ampere, volt, ohm, siemen (mho), joule, watt
Reference directions
Kirchhoff’s Current Law (KCL), Kirchhoff’s Voltage Law (KVL),
Ohm’s Law
Series connection, parallel connection
DC (steady), AC (time-varying)
Independent and dependent ideal voltage and current source
Voltage divider, current divider
Analog (A/D), Digital (D/A)
Multimeter (DMM), Oscilloscope
Prefixes (milli-, etc.)
Linear, nonlinear elements
Superposition (analysis)
Nodal analysis (node, supernode)
Loop analysis (mesh, branch)
Power delivery, dissipation, storage, maximum power transfer
Equivalent circuits (Rs, Cs or Ls in series/parallel; Thevenin,
Norton)
Frequency; angular frequency; period; phase (Hz; radian/s)
Capacitor, inductor, transformer
Phasor, impedance, reactance
Amplifier, filter, transfer function
Steady-state, transient, sinusoidal excitation
Terms2
EECS40, Fall 2004
Lecture 11, Slide 5
Prof. White
Lecture #11
•
•
•
•
•
OUTLINE
The operational amplifier (“op amp”)
Feedback
Comparator circuits
Ideal op amp
Unity-gain voltage follower circuit
Reading
Begin Ch. 14
EECS40, Fall 2004
Lecture 11, Slide 6
Prof. White
The Operational Amplifier
• The operational amplifier (“op amp”) is a
basic building block used in analog circuits.
– Its behavior is modeled using a dependent source.
– When combined with resistors, capacitors, and
inductors, it can perform various useful functions:
•
•
•
•
•
•
•
•
amplification/scaling of an input signal
sign changing (inversion) of an input signal
addition of multiple input signals
subtraction of one input signal from another
integration (over time) of an input signal
differentiation (with respect to time) of an input signal
analog filtering
nonlinear functions like exponential, log, sqrt, etc
EECS40, Fall 2004
Lecture 11, Slide 7
Prof. White
Op Amp Circuit Symbol and Terminals
V+
non-inverting input
inverting input
positive power supply
+
output
–
V – negative power supply
The output voltage can range from V – to V + (“rails”)
The positive and negative power supply voltages
do not have to be equal in magnitude.
EECS40, Fall 2004
Lecture 11, Slide 8
Prof. White
Op Amp Terminal Voltages and Currents
• All voltages are referenced to a common node.
• Current reference directions are into the op amp.
V+
ic+
ip
+
vp
–
+
in
+
vn
–
–
io
+
icV–
–
vo
+
–
Vcc
–
Vcc
+
common node
(external to the op amp)
EECS40, Fall 2004
Lecture 11, Slide 9
Prof. White
Op Amp Voltage Transfer Characteristic
vo
The op amp is a
differentiating amplifier:
Vcc
slope = A >>1
vid = vp–vn
-Vcc
Regions of operation: “negative saturation”
“linear”
“positive saturation”
 In the linear region, vo = A (vp – vn) = A vid
where A is the open-loop gain
 Typically, Vcc  20 V and A > 104
 linear range: -2 mV  vid = (vp – vn)  2 mV
Thus, for an op amp to operate in the linear region,
vp  vn
(i.e., there is a “virtual short” between the input terminals.)
EECS40, Fall 2004
Lecture 11, Slide 10
Prof. White
Achieving a “Virtual Short”
• Recall the voltage transfer characteristic of an op amp:
Plotted using different scales
for vo and vp–vn
Plotted using similar scales
for vo and vp–vn
vo
vo
Vcc
slope = A >>1
~10 V
vp–vn
Vcc
slope = A >>1
~10 V
-Vcc
vp–vn
-Vcc
~1 mV
~10 V
Q: How does a circuit maintain a virtual short at the input of
an op amp, to ensure operation in the linear region?
A: By using negative feedback. A signal is fed back from
the output to the inverting input terminal, effecting a stable
circuit connection. Operation in the linear region
enforces the virtual short circuit.
EECS40, Fall 2004
Lecture 11, Slide 11
Prof. White
Negative vs. Positive Feedback
Familiar examples of negative feedback:
• Thermostat controlling room temperature
• Driver controlling direction of automobile
• Pupil diameter adjustment to light intensity
Familiar examples of positive feedback:
• Microphone “squawk” in sound system
• Mechanical bi-stability in light switches
EECS40, Fall 2004
Lecture 11, Slide 12
Fundamentally
pushes toward
stability
Fundamentally
pushes toward
instability or
bi-stability
Prof. White
Op Amp Operation w/o Negative Feedback
(Comparator Circuits for Analog-to-Digital Signal Conversion)
1. Simple comparator with 1 Volt threshold:
 V– is set to 0 Volts (logic “0”)
 V+ is set to 2 Volts (logic “1”)
 A = 100
+
VIN
V0

V0
2
1
If VIN > 1.01 V,
V0 = 2V = Logic “1”
0
1
1V +

2
VIN
If VIN < 0.99 V,
V0 = 0V = Logic “0”
2. Simple inverter with 1 Volt threshold:
 V– is set to 0 Volts (logic “0”)
 V+ is set to 2 Volts (logic “1”)
 A = 100
+
V0

1V +

V0
2
1
0
1
If VIN < 0.99 V,
V0 = 2V = Logic “1”
2
VIN
If VIN > 1.01 V,
V0 = 0V = Logic “0”
VIN
EECS40, Fall 2004
Lecture 11, Slide 13
Prof. White
Op Amp Circuits with Negative Feedback
Q: How do we know whether an op amp is
operating in the linear region?
A: We don’t, a priori.
• Assume that the op amp is operating in the linear region
and solve for vo in the op-amp circuit.
– If the calculated value of vo is within the range from -Vcc to +Vcc,
then the assumption of linear operation might be valid. We also
need stability – usually assumed for negative feedback.
– If the calculated value of vo is greater than Vcc, then the
assumption of linear operation was invalid, and the op amp
output voltage is saturated at Vcc.
– If the calculated value of vo is less than -Vcc, then the
assumption of linear operation was invalid, and the op amp
output voltage is saturated at -Vcc.
EECS40, Fall 2004
Lecture 11, Slide 14
Prof. White
Op Amp Circuit Model (Linear Region)
vp
Ri is the equivalent resistance “seen” at the
input terminals, typically very large (>1MW),
so that the input current is usually very small:
ip = –in  0
ip
Ro
Ri
+ A(v –v )
p
n
–
vn
in
io
vo
i p  in  io  ic   ic   0
io  (ic   ic  )
Note that significant output current (io)
can flow when ip and in are negligible!
EECS40, Fall 2004
Lecture 11, Slide 15
Prof. White
Ideal Op Amp
• Assumptions:
– Ri is large (105 W)
– A is large (104)
– Ro is small (<100 W)
ip = –in= 0
vp = v n
• Simplified circuit symbol:
– power-supply terminals and
dc power supplies not shown
ip
+
+
in
+
Note: The resistances used
in an op-amp circuit must be
much larger than Ro and
much smaller than Ri in
order for the ideal op amp
equations to be accurate.
EECS40, Fall 2004
vp
–
Lecture 11, Slide 16
vn
–
–
io
+
vo
–
Prof. White
Unity-Gain Voltage Follower Circuit
VIN
vp
+

IIN
vn
V0(V)
V0
vp = vn  V0 = VIN
2
1
1
2
VIN(V)
( valid as long as V –  V0  V + )
Note that the analysis of this simple (but important) circuit required
only one of the ideal op-amp rules.
Q: Why is this circuit important (i.e. what is it good for)?
A: A “weak” source can drive a “heavy” load; in other
words, the source VIN only needs to supply a little power
(since IIN = 0), whereas the output can drive a powerhungry load (with the op-amp providing the power).
EECS40, Fall 2004
Lecture 11, Slide 17
Prof. White