L4 2 4 Inverting AmplifierChap 4

Derivation of the gain of an Inverting Operational Amplifier Circuit

>> The next of the common op-amp configurations

that we're going to consider is known as the inverting amplifier.

It takes its name from the fact that the source voltage that's going to be

amplified is connected to the inverting terminal.

So, with that then,

we might have rightfully assume that because of the name inverting amplifier,

that the output voltage is going to be the opposite in sign of the source voltage.

So, let's go ahead and analyze this as in the same manner that we have

the other op-amp circuits by writing a node equation at the inverting op-amp terminal.

Before we do that we're going to remember that due to the virtual short,

the voltage at the inverting terminal V sub n is

equal to the voltage at the non-inverting terminal,

and in this case, V sub p

the voltage at the non-inverting terminal has been tied to ground.

So, V sub n is going to equal zero also.

Lets go ahead and write the equation leaving V sub n in place so that we can see what's

happening and we'll go back and replace V sub n with zero.

So, starting here, adding the currents leaving

this node starting with the current going in that direction we have,

V sub n minus V sub s divided by R sub s,

plus the current leaving the node going in that direction is going to be

V sub n minus V out divided by R sub f,

plus the current entering the inverting terminal of the op-amp which of course is zero.

So, there's nothing right there,

thus the sum of those two terms must equals zero.

Now, let's replace V sub n with zero in both places.

That's zero. That then is zero and we're left with

negative V sub s over R sub s minus V out over R sub f equals 0.

So, let's just take this negative V out over R sub f to the other side of

the equation as a positive V out over R sub f.

It's a simple task now to solve for V out as

R sub f. Multiplying both sides of the equation by R sub f we get then that

V out is equal to negative V sub s

times R sub f over R sub s. As we anticipated,

the sign on the output is going to be opposite sign of the source voltage,

and we then can note that the gain for

the inverting op-amp is equal to negative R sub f over R sub

s. It's good to compare this gain term with

a gain term that we derived in

the non-inverting amplifier and go back and look at your notes.

But, you may recall that the gain for the non-inverting amplifier was equal to

one plus R sub f over r sub s. This is supposed to be R sub s there.

When you compare these two terms,

you'll notice they're very similar.

They both have the ratio R sub f over R sub s. But in the non-inverting case,

there's also one added to it.

So, for the same circuit, the gain,

if the source was on the non-inverting terminal,

would be one larger than the gain

experienced when the source is on the inverting terminal.

Now, what does that inversion mean?

Well, let's just take a couple of examples here real fast.

Let's let R sub f equal 2 kilo-ohms,

and we'll let R sub s equal 1 kilo ohm.

So, that the ratio R sub f over R sub s is equal to 2,

and for this first instance,

for the first example,

let's just assume that V sub s equals say 5 volts.

Then our output voltage would be V out

would equal negative V sub s

times 2 or negative 10 volts.

So, DC values are just inverted or just have an opposite sign.

What about time-varying voltages?

What if our V source was equal to

say 5 cosine of omega t. What would the output voltage be?

Well, V out would be negative 2 times 5 times the cosine of omega t,

or that's equal to negative 10 cosine of omega t. Well,

what does that sine, what's the amplitude of the output?

Where the amplitude is going to be twice as large as the amplitude of the input,

but it's also going to have a sign reversal. What does that mean?

Well, if this was our original,

where this was five,

the output is going to have twice the amplitude.

Let's just say that that's twice the amplitude.

But it's also going to have a sign reversal.

That means wherever the input was positive,

the output is going to be negative and wherever the input with negative,

the output will be positive and so on.

In other words, we say also,

or another way of putting it is that the output is

180 degrees out of phase with the input and

that's what the minus sign in the inverting op-amp gain term does to us.

Putting it that way makes it sound a little ominous.

What it does to us, now it's not doing anything to us,

it's not hurting us and generally speaking,

that sign inversion doesn't really impact anything.

It just sits there,

and when you build these kinds of circuits up in

the laboratory and look at them on the oscilloscope,

you'll note that there's a 180 degree phase shift.