I'm trying to find an expression for this circuit.

http://www.badongo.com/pic/4698671

The calculations are not very easy using the node method and the golden rules of op-amps. Maybe I can use superposition for V1 and V2? How do I know if this circuit is linear?

 

Its just a subtractor circuit.
Applying KCL at the two input terminals and rearranging the equations would give the output voltage

 

How do I know if this circuit is linear?

That is an incredibly intelligent question!

You should always ask that question because very few things in the real world are truly linear. Also, technically speaking, any active device is highly nonlinear. That means transistors and op-amps are nonlinear, so any linear analysis that is done is an approximation over a limited range of signal amplitudes.

You have to answer your question using judgement. If you can say that your individual components are approximately linear over the amplitude range of interest, then you have a good idea that a linear analysis has some validity.

In your case, you could say that the resistors are linear and the op-amp is linear as long as the output voltage does not hit the rails.

There is a strick mathematical defintion of linearity, but that doesn't help you when you are unsure how to solve the problem. Eventually, you just develop the judgement and know by looking at the circuit.

 

That is an incredibly intelligent question!

You should always ask that question because very few things in the real world are truly linear. Also, technically speaking, any active device is highly nonlinear. That means transistors and op-amps are nonlinear, so any linear analysis that is done is an approximation over a limited range of signal amplitudes.

You have to answer your question using judgement. If you can say that your individual components are approximately linear over the amplitude range of interest, then you have a good idea that a linear analysis has some validity.

In your case, you could say that the resistors are linear and the op-amp is linear as long as the output voltage does not hit the rails.

There is a strick mathematical defintion of linearity, but that doesn't help you when you are unsure how to solve the problem. Eventually, you just develop the judgement and know by looking at the circuit.

Of course, the proof is in the pudding! If you feed a sawtooth or a triangle into the thing and your ramps are straight lines, it's linear!

On a side note.....superposition is such an amazing phenomenon...something we have always taken for granted...but what a strange and different universe we'd live in if superposition didn't apply. It works for electronic, mechanical, even CHEMICAL processes!

eric

 

superposition is such an amazing phenomenon...something we have always taken for granted...but what a strange and different universe we'd live in if superposition didn't apply. It works for electronic, mechanical, even CHEMICAL processes!

I agree. One thing I've always found interesting is that one of the true linear theories is quantum mechanics. One of my physics professors challenged the class to think of a quantum mechanical problem where superposition was not valid. We couldn't think of one, and neither could he. It's easy to find examples in electronics, mechanics, chemistry etc. If anyone can think of an example in QM, I'd like to hear, but please do it as a separate thread so we don't hijack this one.

 

Of course, the proof is in the pudding! If you feed a sawtooth or a triangle into the thing and your ramps are straight lines, it's linear!

You have to be careful here. This property only indicates an "incrementally linear system" which is not strictly a linear system in which superposition applies. Linear systems must have the important property that zero input equals zero output. A system that has a built-in offset does not obey superposition, but can still give straight-lines out with straight lines going in.

You can use the superposition principle for AC signals (around a Q-point) in an incrementally linear system, but not for the true signals.

I'm sure you understand this, but I just want to stress it for boks since he is in the learnig process. A good defintion of linearity is:

if x1 is an input to a system and generates y1 for an output,
and x2 is an input to a system and generates y2 for an output,
then a*x1+b*x2 as an input will generate a*y1+b*y2 for an output,
where a and b are constants.

 

steveb,

...One of my physics professors challenged the class to think of a quantum mechanical problem where superposition was not valid. ...

Would not linear or angular momentum qualify as being nonlinear? After all, they depend of the square of the translational velocity or angular velocity.

Whoops, sorry for the mistake. I was thinking of kinetic energy. Disregard what I said. Apologies to all.

Ratch