Summing op-amp circuit

- [Voiceover] Another form
of an op-amp circuit is

called the summing op-amp.

We're gonna work through
how this one works.

Let's draw in here now is an inverting

op-amp circuit with a single input.

We're gonna call this

this will be V.

We'll call the VA for now.

And we have V out.

And we worked out what
how this worked before

where V out was a function of

VA times the ratio of these two resistors.

And now I'm gonna add
a little twist to this

and we're gonna analyze a
different kind of circuit.

So I'm gonna add another
resistor right here.

Like this.

And a second input.

We'll call this VB.

And so I have RB, or sorry, RA here.

This will be RB.

And we'll call this R feedback.

And the question again is now,

what is V out in terms of

the inputs, VA and VB?

We're gonna use the idea
of a virtual ground.

Virtual ground.

The idea of a virtual
ground applies to most

op-amp circuits, and
it's a really useful way

to simplify the analysis.

And when we use a virtual ground

one of the things I like to do is just

draw a little symbol here

like that, for my virtual ground.

The virtual ground, as a review,

if the voltage coming out of this op-amp

is in a reasonable range,
sort of a plus or minus

10 volts, or something like that.

And because the gain of
this op-amp is so enormous

on the order of 100,000, or a million

that means that, when
this is working properly

that these two voltages will
be really close together.

They'll be only microvolts apart, at most.

And, because they're so close

we can just say, let's
just say they're the same

for the purposes of
analyzing this circuit.

And, since this input is at zero volts

this is at zero volts.

That means that this input
is very close to zero volts.

So this node here

this node here, we say

is at a virtual ground.

So let's move ahead,
let's analyze this circuit

and what we want to
find is we want to find

V out as a function of

VA and VB.

Let's see if we can do that.

So looking at my circuit the other thing

I know about op-amps
that's really important

always is, that the current

going into an op-amp input

is zero, or practically zero.

And for the purposes of
what we're doing here

we can treat it as zero.

All right.

So, we have zero volts on this node.

We have zero current going this way.

So let's go after

let's go after this current right here.

Let's figure out what that is.

So we'll call that I.

And we can express I in terms of VA

and VB in these two resistor values.

So we can write I

equals, now what is it?

Well, we have IA here.

And we have IB flowing here.

That means that I equals

IA plus IB.

All right.

And let me write IA and IB in terms of

Ohms Law here.

So we get I equals, what is IA?

IA is this voltage

divided by RA.

And this voltage is VA

minus zero volts.

So it's VA

over RA.

And now let's write down what's IB?

IB

we look at this voltage here.

This voltage is VB minus

zero.

Or just plain

VB over RB.

All right, so that's one step.

We've written this current in terms of

what's going on on the
input side over here.

Now one thing we know,
because this current is zero,

we know, what?

We know

this current is I.

And, if I's going into RF

we'll label the voltage on RF like that.

So now let's write an expression for I

in terms of RF and this voltage here.

What is this voltage here?

Well, let's look at the two ends.

This is zero volts on this node here.

And on this side it's

V out.

So, the current there, the current

on that side

equals

it's zero minus V out.

So minus V out

divided by RF.

That's Ohms Law for this
resistor right here.

Now we know that this
current equals this current.

So let's just set these
two expressions together.

So I'll do that over here.

I can write minus V out

over RF

equals this one

VA over RA

plus VB over RB.

Now we can get the final function

I'm just gonna multiply
through by RF on both sides.

And we get

V out equals

take the minus sign with us

minus

RF

over RA

times VA

plus

RF

over

RB

times

VB.

So that's our answer.

That's V out

as a function of

VA and VB, the two inputs.

And you can see that the

resistor ratio, there's
two resistor ratios here,

that are participating in the answer.

So it helps to do a little
special case of this.

Let's let all the resistors be the same.

So we'll say that

RA equals RB

equals RF.

And just to pick a real
number we'll just say

they're all equal to 10 K ohms.

And let's see what this becomes now.

Now we have V out

equals, we still have the negative sign.

So this is some sort of
inversion going on here.

RF over RA is one.

So it's just VA.

And RF over RB is one.

So this says

VB.

So this is what gives us the nickname

for this expression

which is called a

a summing

op-amp.

Let's say I want to use my summing op-amp

in an application where what I want is

I want V out to equal say

minus two times VA

plus three times VB.

Okay, this is what I want.

How do I do that?

So, these two coefficients here are

functions of the resistor values.

That was our original expression up here.

So what I want is

RF over RA

to equal two.

And I want RF

over RB

to equal three.

So I can pick values

it's the same RF

and so I get to adjust RA and RB in here.

So I can pick component values.

So if I pick

RF equals 12 K ohms.

Then I can pick

RA equals six K.

And RB

equals four K.

And that would give me the ratios I want

and that would implement

this function here.

So I'm gonna go fill those out

on the top schematic.

We'll just write those in.

And we'll go fill these in here.

So what we said was RF was 12 K.

RA was six K ohms.

RB would be four K ohms.

And we've designed a circuit

that implements our summing function.

So, so this is a pretty
useful op-amp circuit.