L4 2 5 Summing AmplifierChap 4

Derivation of the gain of a Summing Operational Amplifier

>> This configuration is known as a summing amplifier.

We are going to find that it takes

its name from the fact that the output voltage is going to

equal a scaled version of the sum of V_1 and V_2.

To analyze this, we're going to do as we always have and that is write

a node equation at the inverting terminal.

Once again, we have the virtual short here

that tells us that V sub n is equal to V sub p,

and again in this case, the non-inverting terminal is tied to ground.

So, V sub n is going to equal zero.

But again, we'll leave V sub n in there to see what's happening,

and then as we move along we'll replace V sub n with zero.

So, let's sum the currents here at this terminal.

Notice that we now have three branches one, two,

three branches connected to the node in

addition to this branch here coming to the inverting terminal,

which is still once again going to have no current going in there.

So, let's write the equations here,

or the equation here.

The current leaving V sub n going in this path is going to

be V sub n minus V_1 divided by R_1,

plus the current leaving this node going in

this direction is going to be V sub n minus V_2,

divided by R_2, plus the current in the feedback loop,

which is going to be V sub n minus V-out divided by R sub f. As already mentioned,

there is no current going into the amplifier itself,

therefore the sum of those three terms equals zero.

Again, noting that V sub n equals zero in each of these,

we have then negative V_1 over R_1 plus negative V_2

over R_2 equals V-out over R sub f. Now,

if we multiply both sides of the equation by R sub f and factor out,

this minus sign is there,

we're going to have them that V-out is equal to

negative R sub f over R_1 times V_1.

Let's put a bracket around here,

plus R sub f,

over R_2 times V_2.

That's supposed to be a plus sign there.

As advertised, the output then is the sum of

the two input voltages scaled by the ratios of the feedback resistance to either R_1,

in the case of V_1, or the feedback resistor divided by R_2

for the second input voltage.

Now, it'll be pretty obvious that if we want the scaling factor to be the same,

we simply make R_1 equal R_2.

Under those circumstances, the output voltage then would be negative R sub f over,

call it R_1 or call it R_2, I guess so.

Let's just be specific and let's say R_1 equals R_2,

equals R sub s. So,

our gain term then would be R sub f over R sub s times V_1 plus V_2.

So, we have a minus sign again that is

an inverted output due to the fact that

these two sources are connected to the inverting terminal.

With R_1 equaling R_2,

both are amplified by the same amount,

and we end up with a scaled version of the sum of those two.

If we don't want any gain out of it,

we simply make R sub f equal R_1 equal to R_2,

and then the output voltage would be just V_1 plus V_2.