dynamic presentation of key concepts module 5 – part 1 fundamentals of operational amplifiers...

Dave ShattuckUniversity of Houston

© Brooks/Cole Publishing Co.

Dynamic Presentation of Key Concepts

Module 5 – Part 1Fundamentals of Operational

AmplifiersFilename: DPKC_Mod05_Part01.ppt



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



Textbook CoverageThis 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



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.



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

Noninverting InputOutput

Schematic Symbol for the Op Amp


© Brooks/Cole Publishing Co. Op Amps: A Structural Definition





-

+

Inverting Input



This means that at the input, the difference in voltage between the input terminals is used.







-

+

Inverting Input



This means that at the output, the voltage is obtained with respect to a reference, usually called ground.



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.

Complete Schematic Symbol for the Op Amp

-

+

Inverting Input


Positive dcpower supply

Negative dcpower supply



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,•ground and -15[V],•+15[V] and -15[V], or •+15[V] and +5[V].


-

+

Inverting Input






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.


-

+

Inverting Input




Go back to Overview

slide.


© 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



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 AmpIgnoring 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.

-

+

vi

+

-

+

- Avi

v+

v-

vo



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.

Equivalent Circuit for the Op AmpThis 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.

-

+

vi

+

-

+

- Avi

v+

v-

vo



Op Amps: A Functional DefinitionIt 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.

Equivalent Circuit for the Op Amp, for Region Marked in Red

-

+

vi

+

-

+

- Avi

v+

v-

vo

i-

i+

+VDC

-VDC

vo

vi

Saturation

In this region,the op amp

behaves like adependent

source

Saturation

+VDC

-VDC Go back to Overview

slide.



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.

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.

-

+

vi

+

-

v+

v-

vo

i-

i+

+VDC

-VDC



-

+

vi

+

-

v+

v-

vo

i-

i+

+VDC

-VDC

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.



-

+

vi

+

-

v+

v-

vo

i-

i+

+VDC

-VDC

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.

The Two Assumptions





-

+

vi

+

-

v+

v-

vo

i-

i+

+VDC

-VDC

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





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+

vi

+

-

v+

v-

vo

i-

i+

+VDC

-VDC

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.

The Two Assumptions





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+

vi

+

-

v+

v-

vo

i-

i+

+VDC

-VDC

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



Go back to Overview

slide.



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.

Signal Flow Diagram

xo=Axixi A



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 . 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.

xo=Axixi=xs-xf A

Sxs

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+

xf=xo



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.

xo=Axixi=xs-xf A

Sxs

-

+

xf=xo



Negative Feedback – NotesThe feedback amplifier, with a gain of , 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 .

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.

xo=Axixi=xs-xf A

Sxs

-

+

xf=xo



Gain with Negative FeedbackNow, 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

xo=Axixi=xs-xf A

Sxs

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+

xf=xo

where the second equation comes by using the definition of the feedback gain . 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

,osfsi xxxxx

.io Axx

.0xxAx so We can combine terms, then we can divide through by xs, and then by (1+A), to get

.1 A

Ax

x

s

o



Gain with Negative FeedbackThis is the gain with negative feedback,

.1 A

Ax

x

s

o

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 A >> 1. Then,

1 ,A A

and we can use this approximation to simplify the gain with feedback, which we call Af, to

1.o

fs

x AA

x A



Gain with Negative FeedbackThus, the gain with negative feedback, Af, is

The only requirement is that A >> 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.

1.fA




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 negative feedback.

1.fA




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.

1.fA

xo=Axixi=xs-xf A

Sxs

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+

xf=xo

This is what we call the virtual short.



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 solve op amp problems. Go back to

Overview slide.

dynamic presentation of key concepts module 5 – part 1 fundamentals of operational amplifiers...

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