ECE 3455 Electronics

1
ECE 3455 Electronics

• Lecture Notes
• Set 4 -- Version 19
• Operational Amplifiers
• Dr. Dave Shattuck
• Dept. of ECE, Univ. of Houston

2
Amplifier Frequency Response

• We will cover material from Sections 2.1 through
  2.6 from the 5th Edition of the Sedra and Smith
  text.
• It may also be useful for you to consult the
  Nilsson and Riedel Electric Circuits text. The
  material is from Chapter 5 of the 6th or 7th
  Editions.

3
Overview of this Part Fundamentals of
Operational Amplifiers

• In this part, we will cover the following topics
• Basic Operational Amplifier Requirements
• Equivalent Circuit for Operational Amplifiers
• Negative Feedback and What it Does

4
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.

Operational amplifiers are most 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 and design
form.
5
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
  
• Inverting input (which should not be called the
  negative input)
• Noninverting input (which should not be called
  the positive input)
• Output (which can be called the output)

Schematic Symbol for the Op Amp
6
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.

Schematic Symbol for the Op Amp
7
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.

Schematic Symbol for the Op Amp
8
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. 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.

Complete Schematic Symbol for the Op Amp
9
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
• 15V and ground,
• ground and -15V,
• 15V and -15V, or
• 15V and 5V.

Complete Schematic Symbol for the Op Amp
10
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 this course, we will assume that our
  op amps are ideal, and will not use these
  additional terminals.

Complete Schematic Symbol for the Op Amp
11
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
12
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, this module will be accurate
enough for all the problems we consider.
13
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 500,000V,
• the output should be zero volts.

Equivalent Circuit for the Op Amp
This 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.
14
Op Amps A Functional Definition

• The coefficient in the dependent source is the
  gain.

Equivalent Circuit for the Op Amp
The gain A is called the differential gain, and
also called the open loop gain, for reasons that
will become obvious soon.
15
Op Amps A Functional Definition

• Conceptually, the gain A is a function of
  frequency.

Equivalent Circuit for the Op Amp
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.
16
Op Amps A Functional Definition

• 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).

Equivalent Circuit for the Op Amp, for Region
Marked in Red
17
Solving Op-Amp Circuits

• We will use two assumptions for analysis and
  design of op amp circuits where the op amp can be
  considered to be ideal.

The Two Assumptions 1) i- i 0. These
currents are 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.
18
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
  much easier.

The Two Assumptions 1) i- i 0. 2) If
there is negative feedback, then v- v. If
not, the output saturates.
19
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 it happens
  whether negative feedback is present or not.

The Two Assumptions 1) i- i 0. 2) If
there is negative feedback, then v- v. If
not, the output saturates.
20
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
  summing-point 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.
21
Is This Reasonable?

• Many students who are seeing this for the first
  time have 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.

The Two Assumptions 1) i- i 0. 2) If
there is negative feedback, then v- v. If
not, the output saturates.
22
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 better.

The Two Assumptions 1) i- i 0. 2) If
there is negative feedback, then v- v. If
not, the output saturates.
23
Negative Feedback Signal Flow Diagrams

• Engineers have developed a way of looking at
  signals called the signal flow diagram. This is
  not a schematic, and does not represent wire and
  specific components. A line represents a path
  that a signal might follow. The signals can be
  voltages or currents. 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.

Signal Flow Diagram
24
Negative Feedback Signal Flow Diagrams

• Now, lets 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. The signs at
  the junction indicate the signs for the summation.

25
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.

26
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.

27
Gain with Negative Feedback

• Now, lets solve for the gain with negative
  feedback, which is xo/xs. We start by writing an
  equation for the summing junction, taking into
  account the signs, to get

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
We then substitute the first equation into the
second to get
We can combine terms, then we can divide through
by xs, and then by (1Ab), to get
28
Gain with Negative Feedback

• This is the gain with negative feedback

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 gtgt 1. Then,
and we can use this approximation to simplify the
gain with feedback, which we call Af, to
29
Gain with Negative Feedback

• Thus, the gain with negative feedback, Af, is

The only requirement is that Ab gtgt 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.
30
Gain with Negative Feedback

• Thus, the gain with negative feedback, Af, is

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 If there is a signal path between the
output of the op amp, and the inverting input,
there will be negative feedback.
31
Gain with Negative Feedback

• Thus, the gain with negative feedback, Af, is

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.
This is what we call the virtual short.
32
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 how to do
  this in the next part.
• 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
  solve op amp problems.

33
Solving Op Amp Circuits

• We have seen that we can solve op amp circuits by
  using two assumptions

The Two Assumptions 1) i- i 0. 2) If
there is negative feedback, then v- v. If
not, the output saturates.
The key to using these assumptions is being able
to determine whether the op amp has negative
feedback. Remember that we have negative
feedback when a portion of the output is returned
to the input, and subtracted from it.
34
Negative Feedback Identification

• We have seen that we can solve op amp circuits by
  using two assumptions

For ideal op amps, we can assume that the op amp
has negative feedback if there is a signal path
from the output to the inverting input of the op
amp.
The Two Assumptions 1) i- i 0. 2) If
there is negative feedback, then v- v. If
not, the output saturates.
35
Negative Feedback Identification

• Most of the time, this feedback path is provided
  by using a resistor.

For ideal op amps, we can assume that the op amp
has negative feedback if there is a signal path
from the output to the inverting input of the op
amp.
36
Negative Feedback Identification

• In general the rule is this If, when the output
  voltage increases, the voltage at the inverting
  input also increases immediately, then we have
  negative feedback.

For ideal op amps, we can assume that the op amp
has negative feedback if there is a signal path
from the output to the inverting input of the op
amp.
37
Negative Feedback Identification

• In general the rule is this If, when the output
  voltage increases, the voltage at the inverting
  input also increases immediately, then we have
  negative feedback.

For ideal op amps, we can assume that the op amp
has negative feedback if there is a signal path
from the output to the inverting input of the op
amp.
These are two different ways of saying the same
thing. However, for most students this becomes
clearer once we see some examples. We will look
at one example in detail next, and then more
examples after that.
38
Inverting Configuration of the Op Amp

• One of the simplest op amp amplifiers is called
  the inverting configuration of the op amp.

39
Inverting Configuration of the Op Amp

• The inverting configuration is distinguished by
  the feedback resistor, Rf, between the output and
  the inverting input, and the input resistor, Ri,
  between the input voltage and the inverting
  input. The noninverting input is grounded.

40
Inverting Configuration of the Op Amp

• Note that the feedback resistor, Rf, between the
  output and the inverting input, means that we
  have negative feedback.

41
Inverting Configuration of the Op Amp

• Note that the feedback resistor, Rf, between the
  output and the inverting input, means that we
  have negative feedback. Thus, we will have a
  virtual short at the input of the op amp,

42
Gain for the Inverting Configuration

• Lets find the voltage gain, which is the ratio
  of the output voltage vo to the input voltage vi.
  To get this, lets define two currents, ii and
  if.

43
Gain for the Inverting Configuration

• Next, since we know that the voltage v- is zero,
  we can write that the current ii is

44
Gain for the Inverting Configuration

• Following a similar approach, since we know that
  the voltage v- is zero, we can write that the
  current if is

45
Gain for the Inverting Configuration

• Next, by applying KCL at the inverting input
  terminal, we can write

46
Gain for the Inverting Configuration

• Finally, we solve for vo/vi, by dividing both
  sides by vi, and then by multiplying both sides
  by -Rf, and we get

47
Gain for the Inverting Configuration

• This is the result that we were looking for. As
  implied by our analysis of negative feedback, the
  gain is not a function of the op amp gain at all.
  The gain is the ratio of two resistor values,

48
Input Resistance for the Inverting Configuration

• Lets find the input resistance of this
  amplifier, which is defined as the Thevenin
  resistance seen by the source. The source is not
  shown here, but is assumed to be at the input.
  Here, we will take the source as the terminals
  connected to vi.

49
Input Resistance for the Inverting Configuration

• The Thevenin resistance seen by the source will
  be the ratio of vi/ii. We have already solved
  for ii, and found that

50
Output Resistance for the Inverting Configuration

• Lets find the output resistance of this
  amplifier, which is defined as the Thevenin
  resistance seen by the load. The load is not
  shown here, but is assumed to be at the output.
  Here, we will take the load as the terminals
  connected to vo.

51
Output Resistance for the Inverting Configuration

• The Thevenin resistance seen by the load can be
  found by setting all independent sources equal to
  zero, and then applying a test source at the
  output. We do this here, applying a current
  source.

52
Output Resistance for the Inverting Configuration

• Now, we solve for vo/it, which is the output
  resistance, Rout.
• We know that v- 0, due to the presence of
  negative feedback.

53
Output Resistance for the Inverting Configuration

• Now, we solve for vo/it, which is the output
  resistance, Rout.
• We know that v- 0, due to the presence of
  negative feedback. Thus,

54
Output Resistance for the Inverting Configuration

• Now, we solve for vo/it, which is the output
  resistance, Rout.
• We know that i- 0, due our first assumption.
  Thus,

55
Output Resistance for the Inverting Configuration

• Now, we solve for vo/it, which is the output
  resistance, Rout.
• Next, we write KVL around the loop marked with
  a dashed green line. Thus,

56
Output Resistance for the Inverting Configuration

• Now, we solve for vo/it, which is the output
  resistance, Rout.
• Since vo 0, we have

57
Testing the Virtual Short Assumption

• Lets test the results we have obtained, so test
  the virtual short assumption. We found the gain,
  input resistance, and output resistance for
  this configuration. Lets check our approach,
  by going back to the original equivalent
  circuit for the op amp. That is, we replace
  the op amp with a dependent source.

58
Testing the Virtual Short Assumption

• Solving this circuit for the gain, vo/vi, we get

If we take the limit as A goes to infinity, we
get the same answer we had before.
59
Is This Assumption Really Valid?

• This is a good question.
• You can check this by performing the solutions
  with actual values for real op amps. Try an open
  loop gain A of 106, and see how close your
  answers are.
• You can also check this by building an op amp
  circuit, and measuring the actual gain, and other
  parameters. You mightbe surprised by how
  accurate this assumption is.

Go back to Overview slide.