# Op Amp Negative Feedback Algebra

Discussion in 'Math' started by KCHARROIS, Dec 26, 2013.

1. ### KCHARROIS Thread Starter Active Member

Jun 29, 2012
297
1
Hello,

I've been reading Op-Amps and Linear Integrated Circuits by Ramakant and I'm stuck yet again with an algebraic equation to solve the negative feed back gain with non inverting input.

So far I understand I got as far as to this formula below but can't solve for Vo and I don't know where to start to solve for Vo.

Vo = A (Vin - R1*Vo/R1 + Rf)

Thanks

2. ### Shiv123 New Member

Dec 26, 2013
1
1
I'm guessing the formula should be Vo = A (Vin - R1*Vo/R2 + Rf)?

In this case take A to the left by division:
Vo/A = Vin - R1*Vo/R2 + Rf

then take R1*Vo/R2 to left by addition:
Vo/A + R1*Vo/R2 = Vin + Rf

Now, factor out Vo:
Vo(1/A + R1/R2) = Vin + Rf

the take (1/A + R1/R2) to the right by division:
Vo = (Vin + Rf)/(1/A + R1/R2)

there you have it....i hope this is what you wanted.

KCHARROIS likes this.

Jun 29, 2012
297
1
Genius!

Thanks

4. ### KCHARROIS Thread Starter Active Member

Jun 29, 2012
297
1
Never mind I'm not sure where you get the R2 from but its not suppose to be in there.

Sorry... Thanks though

5. ### WBahn Moderator

Mar 31, 2012
20,245
5,758
There are hundreds (probably thousands) of op amp circuits. The equation that applies depends on the circuit. Don't make us guess -- post the circuit.

There's no need, however, to see the circuit to know that the equation you have above is wrong. Look at the units!

(Vin - R1*Vo/R1 + Rf)

This is (volts) - (volts) + (ohms)

Can't be done.

Without the circuit, we can't begin to even guess where you've gone wrong because we are not mind readers.

Well, actually that's not competely true. I think I can guess what you meant the equation to say, but engineering is not about guessing -- that gets people killed. Take the time to express your equations properly and we can go from there.

LvW likes this.
6. ### KCHARROIS Thread Starter Active Member

Jun 29, 2012
297
1
Ok the circuit is like the one attached...

So the book goes like this:

Closed loop voltage gain: Af = Vo/Vin

however: Vo = A(V+ - V-)

But: V+ = Vin and V- = Rg*Vo / Rg+Rf

Therefor putting these formulas together you get: Vo = A(Vin - Rg*Vo/Rg + Rf)

And then solving for Vo you get: Vo = A(Rg + Rf)vin/ Rg + Rf + ARg

I can't personaly solve how they got to the last equation algebraically with

Vo = A(Vin - Rg*Vo/Rg + Rf)???

Thanks

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7. ### #12 Expert

Nov 30, 2010
17,897
9,316
Vo = Vin [1+(Rf/Rg)]

I can't solve their equation either. I think in terms of voltage matching at the inputs, and this works.

8. ### WBahn Moderator

Mar 31, 2012
20,245
5,758
A huge part of your problem is that you are being sloppy, sloppy, sloppy with your math.

V- = Rg*Vo / Rg+Rf

By order of operations -- remember multiplication and division before addition and subtraction? -- this is

V- = ((Rg*Vo) / Rg) + Rf

Which is not what you want. You want

V- = Rg*Vo / (Rg + Rf)

You need to start writing your expressions correctly, otherwise you will forever have serious problems with the algebra.

Because you are being so sloppy with the math and because you are skipping too many steps at once. Slow down and take it stop by step.

Vo = A(V+ - V-)

Vo = A(Vin - (Rg*Vo / (Rg+Rf)))

Vo = A*Vin - A*(Rg*Vo / (Rg+Rf))

Vo + A*(Rg*Vo / (Rg+Rf)) = A*Vin

Vo + Vo *A*(Rg/(Rg+Rf)) = A*Vin

Vo (1 + A*(Rg/(Rg+Rf)) = A*Vin

Af = Vo/Vin = A/(1 + A*(Rg/(Rg+Rf))

Af = A*(Rg+Rf)/((Rg+Rf) + A*Rg)

Af = (Rg+Rf)/((Rg+Rf)/A + Rg)

if A >> (Rg+Rf)

Af ≈ (Rg+Rf)/Rg

Af ≈ 1 + (Rf/Rg)

LvW likes this.
9. ### KCHARROIS Thread Starter Active Member

Jun 29, 2012
297
1
All's I can say is thanks and that yes your right my math is sloppy it does need a lot of work.

10. ### WBahn Moderator

Mar 31, 2012
20,245
5,758
As long as you recognize it and take steps to improve it, you will be fine.

A key part is matching the pace at which you crank through the math to your ability to do so accurately. That's something everyone has to always be on the guard for and I frequently have to stop myself and force myself to slow down and do it carefully and correctly. It's just that, through LOTS of mistakes, I have learned to usually -- usually -- catch when I am running ahead of myself so that I CAN slow down.

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