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>> The next of the common op-amp configurations

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

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

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amplified is connected to the inverting terminal.

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So, with that then,

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

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

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

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

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

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the voltage at the inverting terminal V sub n is

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equal to the voltage at the non-inverting terminal,

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and in this case, V sub p

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

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

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

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

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So, starting here, adding the currents leaving

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

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

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

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

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

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So, there's nothing right there,

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thus the sum of those two terms must equals zero.

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

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

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

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

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

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

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

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V out is equal to negative V sub s

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

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

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and we then can note that the gain for

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

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

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a gain term that we derived in

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

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

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

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When you compare these two terms,

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you'll notice they're very similar.

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

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there's also one added to it.

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So, for the same circuit, the gain,

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if the source was on the non-inverting terminal,

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would be one larger than the gain

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experienced when the source is on the inverting terminal.

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Now, what does that inversion mean?

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

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

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

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

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and for this first instance,

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for the first example,

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

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Then our output voltage would be V out

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would equal negative V sub s

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times 2 or negative 10 volts.

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

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What about time-varying voltages?

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What if our V source was equal to

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

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

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

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

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

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

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Well, if this was our original,

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where this was five,

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the output is going to have twice the amplitude.

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

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

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That means wherever the input was positive,

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

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the output will be positive and so on.

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In other words, we say also,

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or another way of putting it is that the output is

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180 degrees out of phase with the input and

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

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Putting it that way makes it sound a little ominous.

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

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it's not hurting us and generally speaking,

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that sign inversion doesn't really impact anything.

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It just sits there,

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and when you build these kinds of circuits up in

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the laboratory and look at them on the oscilloscope,

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