# summing amplifier equation

Discussion in 'General Electronics Chat' started by Xufyan, Aug 9, 2011.

1. ### Xufyan Thread Starter Member

Aug 3, 2010
114
0
How to derive a general equation for a non-inverting summing amplifier , i have done this for an inverting but stuck for non-inverting as there are resistors on both the inputs ,
what will be the equation of kcl ??

why there is no derivation for non-inverting summing opamp in any book

Last edited: Aug 9, 2011
2. ### steveb Senior Member

Jul 3, 2008
2,433
469
This circuit will behave like a non-inverting amplifier with the V+ voltage multiplied by 1+R4/R3.

The value of V+ is the voltage at the noninverting terminal, which is just (V1 R2+V2 R1)/(R1+R2) which can be easily derived because no current flows into the + terminal.

Now if all resistors are equal, then the gain is 2 and the V+ is (V1+V2)/2 which means the output is V1+V2.

So, the derivation is just an extension of the normal noninverting amplifier derivation, which is probably why you had trouble finding it.

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3. ### ErnieM AAC Fanatic!

Apr 24, 2011
7,907
1,789
Assume: R1=R2=R3=R4 === R

At amp + in (V+):

Think of the inputs as a voltage divider between V1 and V2 to get the first term. Then add in V1 as that's what the difference sits on.

V+ = (V2 - V1) R/(2R) + V1 = (V2 - V1) R/(2R) + V1*2R/(2R)

= (RV2 - RV1 + 2RV1) / (2R) = (RV2 + RV1) / (2R) = (V2 + V1) / 2

VOut = V-/R * R + V- = 2V-

Since V- = V+

VOut = 2(V2 + V1) / 2 = V2 + V1

(Sorry, not in the mood to TEX this)

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4. ### JMac3108 Active Member

Aug 16, 2010
349
67
Whenever you have an op-amp circuit with multiple inputs, the easiest way to solve is usually by superposition. As a reminder, any linear circuit can be solved by superposition. In other words by determining the output from each source individually, then adding up the results. This is done by shorting all the voltage sources except the one you are solving for, then finding the output. Do this for each source, then sum the outputs.

In this circuit there are two sources, V1 and V2.

Step 1 - Short V2 to ground and determine output from V1

The circuit becomes a standard non-inverting amp with a voltage divider consisting of R1 and R2 as the input. The input is V1(R1/(R1+R2)) and this is multiplied by the gain of a non-inverting amp (1+ (R4/R3)).

Vout1 = V1(R1/(R1+R2)) (1+(R4/R3))

Step 2 - Short V1 to ground and determine output from V2

The circuit again becomes a standard non-inverting amp with a voltage divider consisting of R2 and R1 as the input. The input is V2(R2/(R1+R2)) and this is multiplied by the gain of a non-inverting amp (1+ (R4/R3)).

Vout2 = V2(R2/(R1+R2)) (1+(R4/R3))

Step 3 - Sum the outputs

Vout = Vout1 + Vout2

Vout = V1(R1/(R1+R2)) (1+(R4/R3)) + V2(R2/(R1+R2)) (1+(R4/R3))

A little simplification yields,

Vout = [V1(R1/(R1+R2)) + V2(R2/(R1+R2))] (1+(R4/R3))

A common/typical configuration of this circuit:

If you made all the resistors the same value, then the equation collapses to the following.

Vout = [V1(1/2) + V2(1/2)] (2)
Vout = V1 + V2

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5. ### Xufyan Thread Starter Member

Aug 3, 2010
114
0
Thanks alot everyone