Lectures Week 8

1
Lectures Week 8

• The operational amplifier (op amp)
• Feedback
• Comparator circuits
• Ideal op amp
• Unity-gain voltage follower circuit
• Reading
• Ch. 14, Secs. 14.1-14.3, 14.5-14.7 lightly,
• 14.8-14.9, 14.10 lightly

2
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.

3
Op Amp Circuit Symbol and Terminals
V
positive power supply

non-inverting input
output
inverting input
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
(example 0V and 3V DC supplies)
4
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

io
vp
in
vo
vn
ic-
V
Vcc
Vcc
common node (external to the op amp)
5
Op-Amp Voltage Transfer Characteristic
vo
The op-amp is basically a differentiating
amplifier
Vcc
slope A gtgt1
vid vpvn
-Vcc
positive saturation
linear
negative saturation
Regions of operation

• ? In the linear region, vo A (vp vn) A vid
• where A is the open-loop gain
• ? Typically, Vcc ? 20 V and A gt 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.)

6
Achieving a Virtual Short

• Recall the voltage transfer characteristic of an
  op-amp
• 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.

Plotted using different scales for vo and vpvn
Plotted using similar scales for vo and vpvn
vo
vo
Vcc
Vcc
slope A gtgt1
slope A gtgt1
10 V
10 V
vpvn
vpvn
-Vcc
-Vcc
1 mV
10 V
7
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

Fundamentally pushes toward stability
Fundamentally pushes toward instability
or bi-stability
8
Op Amp Operation without 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

• 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

9
Op Amp Circuits with Negative Feedback

• Q How do we know whether an op-amp is operating
  in the linear region?
• A We dont, 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.

10
Op Amp Circuit Model (Linear Region)
Ri is the equivalent resistance seen at the
input terminals, typically very large (gt1MW, as
large as 1012 W for FET input op-amps), so that
the input current is usually very small ip in
? 0
ip
vp
Ro
io
Ri
vo

A(vpvn)
in
vn
Note that significant output current (io) can
flow when ip and in are negligible!
11
Ideal Op-Amp

• Assumptions
• Ri is large (?105 W)
• A is large (?104)
• Ro is small (lt100 W)
• Simplified circuit symbol
• power-supply terminals and
• dc power supplies not shown

ip in 0 vp vn
ip

io
vp
in
vo
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.
vn
12
Unity-Gain Voltage-Follower Circuit
vn
IIN
vp
vp vn ? V0 VIN ( 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 power-hungry load
(with the op-amp providing the power).
13
Whats Inside an Op-Amp?
14
Lecture Week 8 (continued)

• Op-Amp circuits continued examples
• Inverting amplifier circuit
• Summing amplifier circuit
• Non-inverting amplifier circuit
• Differential amplifier circuit
• Current-to-voltage converter circuit
• Reading
• (Note amplifiers are discussed
• in great detail in Ch. 11)

15
Review Negative Feedback
Negative feedback is used to linearize a
high-gain differential amplifier.
With feedback
Without feedback
V0(V)
V0
5
5
4
3
2
1
VIN
1
2
3
4
5
?5
16
Gain vs. Frequency of the Basic Op-Amp without
and with Feedback (Hambley, Sec. 14.5)

• Facts
• The open-loop gain of an op--amp (no feedback
  from output to inverting input)
• is constant from DC to a frequency fBOL, after
  which the open-loop gain drops at a rate of
• -20dB/decade of frequency (Fig. 14.20 next
  slide).
• As the negative feedback is made stronger, the
  gain decreases but the bandwidth
• of the amplifier increases. (Bandwidth of an
  amplifier is the frequency range over which
• it amplifies.)
• One can show (Hambley) that the product of the dc
  gain and the bandwidth
• is a constant that is independent of the amount
  of negative feedback. This is called the
• gain-bandwidth product for the op-amp. Example
  The op-amp youll use in the lab
• (National Semiconductor LMC6482) is rather like
  that shown in Hambley Fig. 14.22,
• with a DC open-loop gain of 100 dB (voltage
  amplification of 105) out to about only
• 40 Hz! With negative feedback, however, the
  amplifier has an increasing bandwidth
• but with decreasing gain. The unity gain (0 dB)
  bandwidth is 4 MHz one can get
• 40 dB of gain (Vout/Vin 100) from DC to about
  40 kHz.

17
Op-Amp Frequency Response with and without
Negative Feedback

18
Application of Voltage Follower Sample
and Hold Circuit
19
Inverting Amplifier Circuit
if
Rf
Rs
in
is

vn
vo

vs
vp
in 0 ? is -if vp 0 ? vn 0
20
Analysis using Realistic Op-Amp Model

• In the analysis on the previous slide, the op-amp
  was assumed to be ideal, i.e.
• Ri ? A ? Ro 0
• In reality, an op-amp has finite Ri, finite A,
  non-zero Ro, and usually is loaded at its output
  terminals with a load resistance RL.

21
Summing Amplifier Circuit
Ra
if
ia
Rf
in
Rb
va

ib

vn
vo
vb

ic
vp
superposition !
Rc
vc

in 0 ? ia ib ic -if vp 0 ? vn 0
22
Application Digital-to-Analog Conversion
A DAC can be used to convert the digital
representation of an audio signal into an analog
voltage that is then used to drive speakers --
so that you can hear it!
Analog
Binary
output
number
(volts)
0
0
0
0
0
0
0
0 1
.5
Weighted-adder D/A converter
S1 closed if LSB 1 S2 " if next bit
1 S3 " if " " 1 S4 "
if MSB 1
(Transistors are used as electronic switches)
LSB
MSB
23
Characteristic of 4-Bit DAC
8
7
6
5
Analog Output (V)
4
3
2
1
0
0
2
4
6
8
10
12
14
16
0000
1111
0001
1000
0100
Digital Input
24
Noninverting Amplifier Circuit
Rf
Rs
in


ip
vo
vn

Rg


vg
vp

ip 0 ? vp vg ? vn vg in 0 ? Rs Rf
form a voltage divider
25
Differential Amplifier Circuit
Ra
Rb
in
vn
va


Rc
ip
vo
vp

vb
Rd
in 0 ? ip 0 ?
26

Differential Amplifier (contd)
More usual version of differential amplifier
(Horowitz Hill, Art of Electronics, 2nd
Ed., p. 184-5 (part of their smorgasbord of
op-amp circuits) Let Ra Rc R1 and Rb Rd
R2 Then v0 (R2 / R1)(vb va) To get good
common-mode rejection (next slide) you need
well-matched resistors (Ra and Rc, Rb and Rd),
such as 100kW 0.01 resistors
27
Differential Amplifier Another Perspective

• Redefine the inputs in terms of two other
  voltages
• 1. differential mode input vdm ? vb va
• 2. common mode input vcm ? (va vb)/2
• so that
• va vcm (vdm/2) and vb vcm (vdm/2)
• Then it can be shown that

common mode gain
differential mode gain
28
Differential Amplifier (contd)

• An ideal differential amplifier amplifies only
  the differential mode portion of the input
  voltage, and eliminates the common mode portion.
• provides immunity to noise (common to both
  inputs)
• If the resistors are not perfectly matched, the
  common mode rejection ratio (CMRR) is finite

29
Op-Amp Current-to-Voltage Converter
30
Summary

• Voltage transfer characteristic of op-amp
• A feedback path between an op-amps output and
  its inverting input can force the op-amp to
  operate in its linear region, where vo A (vp
  vn)
• An ideal op amp has infinite input resistance Ri,
  infinite open-loop gain A, and zero output
  resistance Ro. As a result, the input voltages
  and currents are constrained
• vp vn and ip -in 0

vo
Vcc
slope A gtgt1
10 V
vpvn
-Vcc
1 mV
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Differentiator (from Horowitz Hill)
35
Op-Amp Imperfections
Discussed in Hambley, pp. 651-665 (of interest if
you need to use an op-amp in a practical
application!)
Linear range of operation Input and output
impedances not ideal values gain and bandwidth
limitations (including gain-bandwidth
product) Nonlinear limitations Output voltage
swing (limited by rails and more) output
current limits slew-rate limited (how fast can
the output voltage change 0.5V/ms for the 741
op-amp, to 6000V/ms for high
slew-rate op-amp) full-power bandwidth DC
imperfections Offset current (input currents
dont sum exactly to zero) offset voltage
There are pins (denoted offsets) for inputs to
help control this.