Operational Amplifiers - Differential Amplifiers

Summary

In this educational video, David Williams, an instructor at Okanogan College, explains the workings of a differential amplifier, distinguishing it from a differentiator. He assumes an ideal operational amplifier and uses the superposition principle to demonstrate how the output voltage is proportional to the difference between the two input voltages, V1 and V2. Williams simplifies the concept by showing how the output is affected by each input separately and then combines these effects to reveal the overall operation. He also discusses special cases, such as when all resistors are equal or when specific pairs of resistors are equal, and their impact on the output.

Takeaways

• 🎓 David Williams, an instructor at Okanogan College, introduces the concept of differential amplifiers, emphasizing their distinction from differentiators.
• 🔍 Differential amplifiers are designed to amplify the difference between two input signals, V1 and V2, with the output (V_out) being proportional to this difference.
• 🧠 The explanation assumes an ideal operational amplifier (opamp), characterized by no current flowing into its input terminals and equal voltages at the non-inverting and inverting terminals.
• 🔬 The superposition principle is used to analyze the circuit's behavior by considering one input at a time while grounding the other.
• 🔗 When V2 is grounded (shorted), the circuit behaves as an inverting amplifier, and the output voltage (V_out1) is calculated based on the input V1 and the resistor values.
• 🔄 Reintroducing V2 but grounding V1 allows the calculation of V_out2, which is the output voltage solely due to V2, considering the voltage division across resistors R2 and RG.
• 🔢 The overall output voltage (V_out) is the sum of the individual effects of V1 and V2 on the output, as derived from the resistor ratios and input voltages.
• 💡 In a special case where all resistors are equal, the output voltage is exactly the difference between V2 and V1, simplifying the amplifier's behavior.
• 🔧 Another special configuration is when R1 equals R2 and RF equals RG, but they are not necessarily the same, which results in a gain factor being applied to the difference between the input voltages.
• 📚 The tutorial concludes by reinforcing the understanding of differential amplifiers and hints at further learning in subsequent videos.

Q & A

What is the main difference between a differential amplifier and a differentiator?

A differential amplifier takes the difference between two input signals, whereas a differentiator, also built with op-amps, takes the derivative of an incoming signal.

What assumptions does David Williams make about the op-amp in his explanation?

David Williams assumes that the op-amp is ideal, meaning that the voltage at the non-inverting terminal is equal to the voltage at the inverting terminal, and that no current flows into either terminal.

How does the superposition principle relate to the explanation of the differential amplifier?

The superposition principle is used to analyze the output of the differential amplifier in relation to one input at a time, assuming the other input is grounded, and then combining the effects to understand the overall behavior.

What is the significance of the voltage at the non-inverting terminal being equal to the voltage at the inverting terminal in an ideal op-amp?

In an ideal op-amp, this equality implies that the input impedance is infinite, and thus no current flows into the inputs, which simplifies the analysis of the circuit.

How does the current through R1 relate to the current through RF in the ideal op-amp model?

In the ideal op-amp model, the current through R1 is equal to the current through RF because the inverting terminal is at a virtual ground, and the currents through the feedback network must be conserved.

What is the expression for V_out1 when only V1 is considered in the differential amplifier circuit?

V_out1 is equal to -(RF/R1) * V1, which represents the output voltage due to V1 when V2 is shorted to ground.

How is V_out2 determined when only V2 is considered in the differential amplifier circuit?

V_out2 is determined by the voltage at the non-inverting terminal, which is V2 * RG/(R2 + RG), and is then used to find the output voltage using the relationship between the currents through R1 and RF.

What is the overall expression for V_out in terms of V1 and V2 when both inputs are considered?

The overall V_out is the sum of the individual effects of V1 and V2 on the output, which is V_out = V2 * (RG/(R2 + RG) * (R1 + RF)/R1) - V1 * (RF/R1).

In what special case would the output voltage V_out be exactly the difference between V2 and V1?

The output voltage V_out would be exactly the difference between V2 and V1 if all resistors (R1, R2, RF, RG) are equal in value.

What is the significance of the resistor values in determining the gain of the differential amplifier?

The resistor values determine the gain of the differential amplifier by setting the ratio of the feedback resistor to the input resistor, which can amplify or attenuate the difference between the input voltages.

Outlines

00:00

🔬 Introduction to Differential Amplifiers

David Williams, an instructor in electronic engineering technology at Okanogan College, introduces the concept of differential amplifiers, distinguishing them from differentiators. He clarifies that differential amplifiers are designed to amplify the difference between two input signals, V1 and V2, rather than taking the derivative of a signal. The output voltage (V_out) is proportional to the difference between these inputs. Williams sets the stage for a deeper exploration of the differential amplifier's operation by assuming an ideal operational amplifier (opamp) with no current flowing into its terminals and equal voltages at the non-inverting and inverting terminals. He then uses the superposition principle to analyze the output's relationship to each input individually, with the other input grounded.

05:01

🧮 Analyzing Differential Amplifier Output

In the second paragraph, Williams delves into the mathematical analysis of the differential amplifier's output. He first considers the scenario where V2 is grounded, effectively turning the circuit into a simple inverting amplifier. From this setup, he derives that V_out1 (the output due to V1 alone) is equal to negative RF/R1 times V1. Next, he examines the case where V1 is grounded, focusing on how V_out2 (the output due to V2 alone) is determined. He finds that the voltage at the non-inverting terminal is a function of V2, split between R2 and RG, and uses this to derive an expression for V_out2. By combining the effects of both V1 and V2, Williams presents a comprehensive formula for the overall V_out, which is a function of the resistor ratios and the difference between V2 and V1. He concludes by discussing special cases, such as when all resistors are equal or when R1 equals R2 and RF equals RG, and how these affect the differential amplifier's gain and output.

Keywords

💡Differential Amplifier

A differential amplifier is a type of electronic amplifier that amplifies the difference in voltage between its two inputs. In the context of the video, the instructor explains how a differential amplifier works by taking the difference between two input voltages, V1 and V2, and producing an output voltage proportional to this difference. The differential amplifier is contrasted with differentiators, which are designed to take the derivative of an incoming signal.

💡Opamp

An operational amplifier, or opamp, is a type of amplifier that is widely used in analog circuits. It is characterized by high gain and is often used in conjunction with feedback networks. In the video script, the opamp is assumed to be ideal, which simplifies the analysis by assuming that no current flows into its input terminals and that the voltages at the non-inverting and inverting terminals are equal.

💡Ideal Opamp Assumptions

The ideal opamp assumptions are a set of simplifications used in circuit analysis to make the calculations more manageable. These include the assumptions that the opamp has infinite gain, infinite input impedance, zero output impedance, and that no current flows into its input terminals. In the video, these assumptions are used to derive the relationship between the input voltages and the output voltage of the differential amplifier.

💡Superposition Principle

The superposition principle is a fundamental concept in circuit analysis that allows for the simplification of complex circuits by considering the effects of each voltage source independently. In the video, the instructor uses the superposition principle to analyze the differential amplifier's output when one input is grounded, and then when the other input is grounded, to determine how the output is related to each input voltage.

💡Non-Inverting Terminal

The non-inverting terminal of an opamp is one of its input terminals, typically labeled as the '+' terminal. It is characterized by a high impedance input, meaning it draws very little current. In the video, the non-inverting terminal is discussed in the context of the ideal opamp assumption that its voltage is equal to the voltage at the inverting terminal.

💡Inverting Terminal

The inverting terminal of an opamp is another input terminal, usually labeled as the '-' terminal. It is also a high impedance input. The video explains that, in an ideal opamp, the voltage at the inverting terminal is equal to the voltage at the non-inverting terminal, which is a key assumption used to simplify the analysis of the differential amplifier.

💡Virtual Ground

A virtual ground is a point in a circuit that is not actually connected to the physical ground but behaves as if it were due to the action of an opamp. It is a concept used to simplify the analysis of circuits with negative feedback. In the video, the instructor refers to the inverting terminal as a virtual ground when analyzing the differential amplifier's behavior.

💡Resistor Ratios

Resistor ratios are the relative values of resistors in a circuit, which play a crucial role in determining the gain and output voltage of an amplifier. In the video, the instructor discusses how the output voltage of the differential amplifier is dependent on the ratios of the resistors R1, R2, RF, and RG, and how these ratios affect the difference in voltage between the two inputs.

💡Feedback

Feedback in an electronic circuit is a process where a portion of the output signal is returned to the input, either to enhance or diminish the original signal. In the video, negative feedback is used in the differential amplifier to stabilize the operation and improve linearity. The feedback network is created by the resistors connected to the output and input terminals of the opamp.

💡Voltage Divider

A voltage divider is a circuit configuration that consists of two or more resistors connected in series, which divides the input voltage across the resistors according to their values. In the video, the voltage divider principle is used to determine the voltage at the inverting terminal of the opamp, which is crucial for understanding how the differential amplifier operates.

Highlights

Introduction to differential amplifiers and their distinction from differentiators.

Explanation of how differential amplifiers output is proportional to the difference between V2 and V1.

Assumption of an ideal opamp with no current flowing into its terminals.

Use of the superposition principle to analyze the output in relation to the inputs.

Analysis of the circuit when V2 is grounded to find the output V_out1.

Derivation of V_out1 formula as V_out1 = -RF/R1 * V1.

Reintroduction of V2 and shorting of V1 to find V_out2.

Calculation of voltage at the non-inverting terminal based on V2.

Derivation of V_out2 formula considering the voltage divider principle.

Final expression for V_out as a function of the difference between V2 and V1.

Special case analysis when all resistors are equal, resulting in V_out being the difference between V2 and V1.

Another special case where R1 equals R2 and RF equals RG, leading to a gain applied to the input difference.

Practical application of differential amplifiers in electronic engineering.

Educational approach to understanding differential amplifiers through step-by-step analysis.

Importance of ideal opamp assumptions in simplifying circuit analysis.

Conclusion and anticipation for the next video in the series.

Transcripts

hi there I'm David Williams I'm an

instructor in the electronic engineering

technology department at Okanogan

college and I want to show you today how

differential amplifiers work now don't

confuse differential amplifiers with

differentiators which can also be built

with opamps differentiators take the

derivative of an incoming signal whereas

these differential amplifiers are going

to take the difference or there the

output anyway is going to be the output

here V out is going to be proportional

to the difference between V2 and V1 and

what I want to show you is how that

relationship comes about so here's my

circuit again and in order to show how

the output is proportional to the

difference between V2 and V1 I'm going

to make a couple of assumptions about

this opamp and and basically I'm going

to assume that this opamp is ideal so

what those assumptions mean or what an

ideal opamp means especially or

considering this one that has a feedback

going from the output back to the in

inverting

input the voltage at the non-inverting

terminal so that's the voltage here with

respect to ground is going to be equal

to the voltage at the inverting terminal

so that's the voltage here with respect

to

ground and the other assumption that I'm

going to need to make use of is that no

current flows into either one of these

terminals no current into the inverting

terminal no current into the

non-inverting terminal and so what that

means is the current through R1

there

is going to be equal to the current

through this RF

here it also means that the current

through this R2 resistor is going to be

equal to the current through the RG

resistor so I'm going to have these

three conditions imposed on The Circuit

by assuming that my opamp here is

ideal now in order to see how the output

relates to the two input over here we're

going to have to make use of one way to

go about doing this is to make use of

the superposition principle so what

we're going to do is see what the output

is in relationship to one of the inputs

assuming that the other input is

grounded and then do the same thing for

the other input see how the output

output is in relationship to V1 here

when V2 V2 is is zero and then when we

put the two of the sum of those two

signals together the sum of those two

voltages together we'll get an overall

picture of what's what's going on in the

circuit so the first thing I'm going to

do is just look at the output look at

the output when is just V1 so I'm going

to short V2 so that means this voltage

here at V2 is grounded is ground and

so what is the output when it's only V

only considering

V1 so in this case if we short V2 here

well that's going to end up making the

non-inverting terminal ground so this

point is is ground and we end up with

just the just an inverting an inverting

amplifier and so we will

have ir1 is equal to I

RF since this point is is virtual ground

ir1 will be V1 over

R1 and again since this is virtual

ground the voltage across RF will be 0

minus V out which is just negative V

out / by

RF and we can rearrange this equation

and we get V

out is equal

to RF over

R1 time V1 so I guess we should call

this V

out1 because this is just the V out due

to V1 so V out1 is equal to negative RF

over R1 * V1

now what if we reintroduce V2 but we

short V1 so now V1 is connected to

ground

here so we short

V1 and what we want to do is find out

what V out is when V when V1 is shorted

so we'll call this V out2 this is just

when V out is dependent only only on V2

so now what we want to do is figure out

what V out2 is when when it's just V1

when when V1 is shorted and it's just V2

here being

applied so again the inverting terminal

and the non-inverting terminal are going

to be the at the same voltage so we

could actually figure out what the

voltage at this point is in terms of V2

so at the non-inverting terminal that

voltage is equal to V2 * RG over R2 plus

RG right because V2 is across both R2

and RG and so it's going to be split

between R2 and RG and the proportion

that split depends on the values of

those two resistors so v v voltage at

the non-inverting terminal terminal will

be equal to V2 *

RG over R2 +

RG and since we have this negative

feedback here the voltage at the

inverting terminal is going to be equal

to the voltage at the non inverting

terminal now we know the voltage at this

point and so we can use the fact that we

will have current going through R1 and

RF those two currents are going to be

equal so we can find out V out in we can

find out the voltage at this point in

terms of a v

out now we can find the voltage at the

inverting terminal in terms of V out

because this voltage is based on what

the V out is and it's based on the

voltage divider between r R1 and RF so V

at voltage at the inverting terminal is

equal to V

out * R1 over R1 +

RF so we've got one expression here for

the inverting terminal voltage and

another expression here for the voltage

at the inverting terminal so we can set

those two equal to each other we'll have

V2 * RG over r2+

RG is equal to

V out * R1 over R1 +

RF now the trick is we want to we want

just have this expression in terms of V

out and I guess this is the V out2

because only due to the voltage from

from uh

V2 so just rearranging this expression V

out2 is equal to

V2 *

rg/ r R2 +

RG * R1 +

RF over R1 so there's my expression for

V

out2 now the overall V

out is equal to the voltage due to V2

voltage at the output due to V2 which is

this expression plus the voltage at V

out due to V1 which is this expression

and so we get V out is equal to

V2 times all of these resistor ratios

here RG over R2 +

RG time R1 +

RF over

R1

minus V1 * RF over

R1 now this is may look like a really

big expression but you can see here that

I mean it if these if these resistors

are set that's just a number that's just

a number multiplying V2 so it's some

some number times V2 minus again RF over

R1 that's just going to be some number

based on the resistors some number times

V1 so V out is based on the some number

time V2 minus some number time V1 so

it's proportional essentially

proportional to the difference between

V2 and

V1 now in the special case where all of

these resistors are equal to each other

so we've got R1 is equal to R2 is equal

to RF is equal to

RG then you'll see this will be 1 over 1

over 2 * 2 over 1 so that will just be

equal to one that this big resistor

expression will be equal to one and this

resistor expression will also be equal

to one so if all those resistors are

equal to each other then the output

voltage will be just the difference

between V2 and

V1 so there's the differential

amplifier another special

case is if R1 is equal to R2 and at the

same time RF is equal to RG but R2 and

RF are not necessarily the same values

so these these this would be one value

and these two resistors would be another

value what we would get is V

out is equal to

some gain which is RF over R1 times the

difference between V2 and V1 so you can

do the diff to do the difference between

those two input voltages but then also

apply some amount of gain based on based

on this

ratio so hopefully you learned a little

bit about differential amplifiers and

I'll see you in the next

video