Download DPKC_Mod05_Part01_v08

Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Dynamic Presentation of Key
Concepts
Module 5 – Part 1
Fundamentals of Operational
Amplifiers
Filename: DPKC_Mod05_Part01.ppt
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Overview of this Part
Fundamentals of Operational Amplifiers
In this part of Module 5, we will cover the
following topics:
• Operational Amplifiers – Structure and
Requirements
• Equivalent Circuit for Operational
Amplifiers – Functional Definition
• Operational Amplifier Circuit Assumptions
• Negative Feedback and What it Does
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Textbook Coverage
This material is introduced in different ways in different
textbooks. Approximately this same material is covered in
your textbook in the following sections:
• Circuits by Carlson: Section 3.3
• Electric Circuits 6th Ed. by Nilsson and Riedel: Sections
5.1 & 5.2
• Basic Engineering Circuit Analysis 6th Ed. by Irwin and
Wu: Section 3.3
• Fundamentals of Electric Circuits by Alexander and
Sadiku: Sections 5.1 through 5.3
• Introduction to Electric Circuits 2nd Ed. by Dorf: Sections
6.1 through 6.4
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Operational Amplifiers (Op Amps)
Operational Amplifiers
(op amps) are devices that
amplify voltages. Because
of the way the op amps are
built, they facilitate the
application of negative
feedback, which in turn
allows
• easy design of special
applications, and
• for the op amps to behave
very ideally.
We will define negative
feedback shortly.
Op amps are useful because of their ability to
process signals easily. One example: An audio
mixer provides the functions of providing
variable gain (amplification), and adding
different signals together with separate gains for
each signal. Op amps can perform these
functions, and do so in an easy-to-use form that
makes designing easier.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Structural Definition
An op amp is a differential input, single-ended output, amplifier.
The schematic symbol for the op amp has three terminals that are
always shown:
1. Inverting input (which should not be called the negative input)
2. Noninverting input (which should not be called the positive input)
3. Output (which can be called the output)
Inverting Input
Output
Noninverting Input
+
Schematic Symbol for the Op Amp
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Structural Definition
This means that
at the input, the
difference in
voltage between
the input
terminals is
used.
An op amp is a differential input, single-ended
output, amplifier. The schematic symbol for the op amp
has three terminals that are always shown:
1. Inverting input (which should not be called the
negative input)
2. Noninverting input (which should not be called the
positive input)
3. Output (which can be called the output)
Inverting Input
Output
Noninverting Input
+
Schematic Symbol for the Op Amp
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Structural Definition
This means that
at the output, the
voltage is
obtained with
respect to a
reference,
usually called
ground.
An op amp is a differential input, single-ended
output, amplifier. The schematic symbol for the op amp
has three terminals that are always shown:
1. Inverting input (which should not be called the
negative input)
2. Noninverting input (which should not be called the
positive input)
3. Output (which can be called the output)
Inverting Input
Output
Noninverting Input
+
Schematic Symbol for the Op Amp
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Structural Definition
There are actually five terminals which are always present in an op
amp. These are shown in the more complete schematic shown below.
The dc power supplies must be connected for the op amp to work. In
some cases they may not be shown, since they do not affect the signal
behavior in many cases. However, the connections must be present,
whether they are shown in the schematic
or not.
Positive
dc
power supply
Inverting Input
Output
Noninverting Input
+
Negative dc
power supply
Complete Schematic Symbol for the Op Amp
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Structural Definition
It is important to note that, in the positive and negative dc power
supplies, the “positive” and “negative” here are relative. The voltages for
these terminals may have any relationship to ground. For example, the
voltages could be:
• +15[V] and ground,
Positive dc
• ground and -15[V],
power supply
• +15[V] and -15[V], or
• +15[V] and +5[V].
Inverting Input
Output
Noninverting Input
+
Negative dc
power supply
Complete Schematic Symbol for the Op Amp
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Structural Definition
Actually, most op amps have at least two more terminals for use in
correcting for some of the non-ideal characteristics of the op amp.
However, for these modules, we will assume that our op amps are ideal,
and will not use these additional terminals.
Positive dc
power supply
Inverting Input
Output
Noninverting Input
+
Negative dc
power supply
Complete Schematic Symbol for the Op Amp
Go back to
Overview
slide.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Functional Definition
Op amps take the difference between the voltages at the two inputs,
and amplify it by a large amount, and provide that voltage at the output
with respect to ground. This can be shown with an equivalent circuit.
Equivalent Circuit for
the Op Amp
+
vi
+
+
Avi
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Functional Definition
The effective resistance between the input terminals, and the resistances
between the input terminals and ground, are typically large compared to
other resistances in the op-amp circuits, and can often be large enough to
be considered effectively infinite. The output resistance is small enough to
be ignored.
Equivalent Circuit for the Op Amp
Ignoring all these resistances
gives us the relatively simple
equivalent circuit at right. In
some situations, these
resistances can be estimated
and included in the equivalent
circuit to make it more accurate.
For the purposes of these
modules, the equivalent circuit
shown here will be accurate
enough for all the problems we
consider.
v-
-
+
-
v+
vi
+
+
Avi
vo
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps:
A Functional Definition
Note that the output is a function of only the difference between
the inputs. This means that if
v- = v+ = 500,000[V],
the output vO should be zero volts.
This property is hard to obtain in
practice, and is called common
mode rejection. The part of the
two voltages, v- and v+, that are
common, is called the common
mode. The gain A is called the
differential gain, and also called
the open loop gain, for reasons
that will become obvious soon.
The response is good for all
frequencies, even at dc, in the
ideal case. The value for the
gain, A, is very large, in general.
Equivalent Circuit for the Op Amp
v-
-
+
-
v+
vi
+
+
Avi
vo
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Op Amps: A Functional Definition
It should also be noted that the output voltage is limited. The output voltage
cannot be higher than the positive dc power supply voltage (+VDC), and cannot be
lower than the negative dc power supply voltage (-VDC). If the output would be
predicted by the equivalent circuit to go higher or lower than these limits, the output
instead will be held at these levels. This is called saturation.
vo
Equivalent Circuit for the Op Amp,
for Region Marked in Red
Saturation
+VDC
+VDC
iv-
-
vi
-VDC
Saturation
In this region,
the op amp
behaves like a
dependent
source
+
i+
v+
Avi
vo
vi
+
+
-VDC
Go back to
Overview
slide.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Solving Op-Amp Circuits
The equivalent circuit in the last slide, combined with an
understanding of saturation, is sufficient to solve any problems we will
encounter in this module. In a sense, this equivalent circuit defines an ideal
op amp. However, in practice we use two assumptions for analysis and
design of op amp circuits where the op amp can be considered to be ideal.
+VDC
The Two Assumptions
1) i- = i+ = 0. These are defined as
the currents into the signal inputs,
and are very small due to the high
input resistances.
2) If there is negative feedback,
then v- = v+. If there is no negative
feedback, the op amp output will
saturate. If vi is positive, it
saturates at +VDC, and if vi is
negative, it saturates at –VDC.
iv-
vi
vo
+ i+
v+
+
-VDC
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Solving Op Amp Circuits
We use these two assumptions for the analysis and design
of op amp circuits where the op amp can be considered to be
ideal. While the equivalent circuit may seem to be easier, we
will see that these assumptions make solving op amp circuits
+VDC
much easier.
The Two Assumptions
1) i- = i+ = 0.
2) If there is negative
feedback, then v- = v+.
If not, the output
saturates.
iv-
vi
vo
+ i+
v+
+
-VDC
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
First Assumption
The first assumption results from having large resistances
at the input, larger than the resistance values typically
connected to them. This assumption is not conditional; the
currents are negligible whether negative feedback is present
or not.
+V
DC
The Two Assumptions
1) i- = i+ = 0.
2) If there is negative
feedback, then v- = v+.
If not, the output
saturates.
iv-
vi
vo
+ i+
v+
+
-VDC
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Second Assumption
The second assumption results from negative feedback and the very
large gain of the op amp. This is called the virtual short, or the summingpoint constraint. The two input voltages are constrained to be equal by the
presence of negative feedback. Without negative feedback, even a small
input will saturate the output.
The Two Assumptions
1) i- = i+ = 0.
2) If there is negative
feedback, then v- = v+.
If not, the output
saturates.
+VDC
iv-
vi
vo
+ i+
v+
+
-VDC
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Is This Reasonable?
Many students who are seeing this for the first time have relatively little
trouble accepting the first assumption. It seems reasonable to be able to
have large input resistances. However, the notion that the input voltage vi
will be forced to zero by something called negative feedback is harder to
accept. Some of these students are troubled by the notion that the input
would be zero, which is then multiplied by a very large number to get a
finite, nonzero output.
+VDC
The Two Assumptions
1) i- = i+ = 0.
2) If there is negative
feedback, then v- = v+.
If not, the output
saturates.
iv-
vi
vo
+ i+
v+
+
-VDC
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Is This Reasonable? Yes!
The notion that the input voltage vi is forced to zero by something
called negative feedback is an approximation. It is actually forced to be
very small, because the gain is so large. Thus, a very small input, which is
almost zero, is then multiplied by a very large number, to get a finite,
nonzero output.
To understand this better, we need to understand negative feedback
+VDC
better.
The Two Assumptions
1) i- = i+ = 0.
2) If there is negative
feedback, then v- = v+.
If not, the output
saturates.
iv-
vi
vo
+ i+
v+
+
-VDC
Go back to
Overview
slide.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Negative Feedback – Signal Flow
Diagrams
To understand negative feedback better, engineers have developed a
way of looking at signals called the signal flow diagram. A signal flow
diagram is not a circuit schematic, and does not represent wires and
specific components. In this, a line represents a path that a signal might
follow. The signals can be voltages or currents; it does not matter.
Therefore, we will label the signals with the symbol x.
In the signal flow diagram shown below, there is an input signal, xi.
This signal flows into an amplifier with gain A, which is shown with a
triangle. This produces an output signal xo. The input is multiplied by the
gain, to give the output.
xi
A
xo=Axi
Signal Flow Diagram
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Negative Feedback – Signal Flow
Diagrams
Now, let’s add negative feedback to our signal flow diagram.
In the signal flow diagram shown below, we add another amplifier. This
amplifier has a gain which is conventionally called b. This amplifier amplifies the
output signal, to produce a feedback signal, xf.
Finally, this feedback signal is subtracted from the input signal. The symbol
for this action is called a summing point or a summing junction. Note that the
signs at the junction indicate the signs for the summation.
xs
+
xi=xs-xf
S
xo=Axi
A
xf=bxo
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Negative Feedback – Signal Flow
Diagrams
Now, let’s add negative feedback to our signal flow diagram.
In the signal flow diagram shown below, we add another amplifier. This
amplifier has a gain which is conventionally called b. This amplifier amplifies the
output signal, to produce a feedback signal, xf.
Finally, this feedback signal is subtracted from the input signal. The symbol
for this action is called a summing point or a summing junction. Note that the
signs at the junction indicate the signs for the summation.
xs
+
xi=xs-xf
S
xo=Axi
A
xf=bxo
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Negative Feedback – Definition
At this point, we can define negative feedback. Negative
feedback is when a portion of the output is taken, returned to the
input, and subtracted from this input.
If we were to add it to the input, we would call it positive
feedback.
xs
+
xi=xs-xf
S
xo=Axi
A
xf=bxo
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Negative Feedback – Notes
The feedback amplifier, with a gain of b, is typically not an
amplifier per se, but rather is a resistive network. In any case, the
key is that the feedback signal xf is proportional to the output signal,
with a multiplier equal to b.
The gain A is called the open loop gain, because this would be
the gain if the loop were to be opened, that is, if the feedback were
removed.
xs
+
xi=xs-xf
S
xo=Axi
A
xf=bxo
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Gain with Negative Feedback
Now, let’s solve for the gain with negative feedback, which is xo/xs. After
these definitions, this is simply an algebra problem. We start by writing an
equation for the summing junction, taking into account the signs, to get
xi  xs  x f  xs  bxo ,
where the second equation comes by using the definition of the feedback gain
b. Next, we use a similar definition for the feed-forward gain, A, to write
xo  Axi .
We then substitute the first equation into the second to get
xo  A xs  bx0 .
We can combine terms,
then we can divide through
by xs, and then by (1+Ab),
to get
xo
A

.
xs 1  Ab
xs
+
xi=xs-xf
S
xo=Axi
A
xf=bxo
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Gain with Negative Feedback
This is the gain with negative feedback,
xo
A

.
xs 1  Ab
If we take the case where A is very large, and it usually is, we can
get a special situation. Specifically, take the case where Ab >> 1.
Then,
1  Ab  Ab ,
and we can use this approximation to simplify the gain with
feedback, which we call Af, to
xo
A
1
 Af 
 .
xs
Ab b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Gain with Negative Feedback
Thus, the gain with negative feedback, Af, is
Af 
1
b
.
The only requirement is that Ab >> 1. Thus, the gain is not a
function of A at all!?! This is a seemingly bizarre, but
wondrous result, which is fundamental to the power of
negative feedback. The gain of the op amp, which changes
from time to time, and from op amp to op amp, does not affect
the overall gain with feedback.
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Gain with Negative Feedback
Thus, the gain with negative feedback, Af, is
Af 
1
b
.
Thus, the gain is not a function of A at all!?! The gain of the op
amp does not affect the overall gain with feedback. The
overall gain, Af, is determined by the way feedback is applied.
Feedback is used to allow gain to be traded off for a
variety of desirable results. When we use op amps, we have
a relatively simple way to determine the presence of negative
feedback. Our rule: If there is a signal path between the
output of the op amp, and the inverting input, there will be
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
Gain with Negative Feedback
Thus, the gain with negative feedback, Af, is
Af 
1
b
.
With this result, we can look again at the signal flow diagram.
The input to the op amp, vi, is the output divided by the gain,
vo/A. If A is large, then vi will be much less than vo, and can
usually be neglected.
xs
xi=xs-xf
xo=Axi This is
+
S
A
what we
call the
virtual
xf=bxo
short.
b
Dave Shattuck
University of Houston
© Brooks/Cole Publishing Co.
How do we use this?
• This is a good question.
• We will use the two assumptions to solve op amp
circuits more quickly. We will show examples of
how to do this in the next part, DPKC Module 5
Part 2.
• The point to recognize here is that negative
feedback can be very useful, and makes op amps
circuits much easier to analyze, and therefore much
easier to design with. Most of the circuits that we
look at will have negative feedback. However, to
prepare for the future, we will always
check for negative feedback when we
Go back to
solve op amp problems.
Overview
slide.