linearization of non-linear systems

[steveB]

OK, so Q1, Q2, Q3 and Q4 are addressed by my opinion above. Whether or not my opinion is accepted is a separate issue, but that's how I will answer those questions. I'll be the first to acknowledge that offset nonlinearities are the very simplest to deal with, and if any nonlinear system can be called a linear system, it would be that one. But, it is important to make the distinction so that one remembers that there is an extra step (albeit a very simple step) that must be added to the analysis to make the system "properly linear".

So, I can try the remaining questions now.

Q5: A resistor is a linear device because the relationship between current and voltage is linear - when voltage is increased, the current also increases proportionally. How would you define linearity of capacitor and inductor along the same lines in DC analysis?

A resistor does have a linear VI curve, but is that why it is called a linear device? I'm not sure because it obeys the linear function rule and the linear system rule. Is it one, the other, or both? I don't know.

Caps and inductors don't obey a linear function rule, but are linear devices in the sense that the VI relationship can be viewed as a system that obeys linearity in the system sense. Also, when these devices are used with other linear devices (most notably resistors) we usually end up with a linear system that has a linear differential equation (or linear state space) representation.

Q6: Almost every circuit has two kinds of responses, transient response and steady state response, where transient response dies away as the time tends to infinity. If it is said that a circuit or electrical system is a linear system, it means that the relationship between input and output is of linear nature. Let's focus on an RC circuit in zero-state with a DC power supply. You can see here that it is a linear system even in view of its transient response in addition to being a linear system in steady state. If an electrical system is a linear one considering its steady state response, then does this mean that it is also linear system when its transient response is considered?

The transient response just adds to the forced response, so the transient response does not really matter for linearity. The reason for excluding the response to initial conditions is that it will mask the response of the system to its input signals. If you dont' separate those responses, you might think the initial condition response is part of your forced response, and so you might think the system is nonlinear because the output does not match what you expect. The basic thing to do is to look at the differential equation, or state space system to see if it is linear. Then you dont' need to worry about this at all. The fact that you can make a linear differential equation (weather time invariant or time varying) makes the linearity clear.

Q7: For an RC circuit the transient response is given as Vout(t)=Vs{1-e^(-t/RC)} where Vs is a DC supply. Let's take R=0, then Vout(t)=Vs{1-1}=0. Could you please tell me that why this is so?

I believe that MrAl answered this well enough.

 

[PG1995]

Thanks a lot, Steve.

Whether or not my opinion is accepted is a separate issue, but that's how I will answer those questions.

I just wanted to clear one thing. No one was trying to negate your opinion (though it might look like this to someone). It was just that this is how we learn. From your side, perhaps, we were just debating one issue of linearity vs. non-linearity. But on my side, I learned many things by debating the matter like how engineering works, how to apply theory correctly, etc. Even if someday I'm able to negate some opinion of yours (which is highly improbably because chimps can never be as smart as humans no matter how much you teach and train them!), you should be proud of it because then as a mentor you can be sure that your hard work, guidance and time didn't go down the drain, and you were able to teach a chimp where many had failed! Thank you.

Best regards
PG

 

[MrAl]

Hi Mr Al,

Those are good points. However, I find your wordings in this thread much less objectionable than what PG is trying to say. You can note that in all of your discussion here, you use the term "linear" and not "linear system". To me there is enough leeway when you don't add the word "system". "Linear" can mean different things in different contexts, whereas "linear system" usually has special meaning to electrical engineers. Only you know if your word choice was conscious or unconscious, but either way, your experience and knowledge leads you to not say "linear system" for a system that does not meet the definition we all use and accept.

But, as I said, we can use any terms we want to, but we should be clear about meanings in serious work. Obviously casual conversation is less critical. In your buck converter example, you are quite clear on what you are saying and doing. You have a nonlinear circuit, and you describe a process of developing an average model. Averaging switch system models do not always yield a linear system (it depends on the exact converter type), but often they do , or when they don't, they can often be approximated as a linear system, particularly for the common converter topologies. For some topologies, that does not work well and you have to linearize the nonlinear averaged model and consider different operating points in your analysis.

The buck converter example really highlights the importance of "linear system" analysis. We are able to take a circuit that would be very hard to model and develop control algorithms for, and we convert it into a linear system model that is "good enough" to allow all the linear analysis and linear control theory to be applied to the problem.

Hi Steve,


Yes i agree with that, except maybe for thinking about the linearization as being "good enough" when it is actually more than good enough, it can be very accurate when applied correctly. But i dont think that is that important for this discussion so i'll skip it and get back to the main issue

I think i see what is happening here after reading your post.

All along i was more or less trying to take the emphasis off of what we call linear and what we dont call linear, while all along you have been trying to put emphasis on the importance. PG is a little mixed up because we're calling the same circuit linear and non linear, so which is it.

I think the reason why is because you would like to see us define linear according to a system that is either linear or not linear because in many systems that classification is certainly very important, while i on the other hand would like to leave it up to the discretion of the reader but i dont mean a random judgment either or one that is clearly contrary to simple reasoning.

But i think you are coming from a linear vs non linear control system point of view, while i was coming more or less from a circuits point of view. For example, if we find a book:
"Linear Control Systems"

what will we expect to find in this book?

I would expect to find only linear control systems with no non linear systems, but i would expect to find op amp circuits that by the strict definition non linear as well as linear. We might also find linear control systems used in non linear systems where we assume a linear operation over a short range of operation. I think the inverted pendulum is an example.

So for me it's that i dont want to be specific sometimes, but other times i do want to be very specific because the whole approach depends on it. So i think what we call linear and non linear depends on the context of what we are looking at and how we approach it and even for selecting an approach.

For the op amp circuit with offset, i think we have to call it non linear but can be used as linear in some applications.

 

[steveB]

Yes i agree with that, except maybe for thinking about the linearization as being "good enough" when it is actually more than good enough, it can be very accurate when applied correctly. But i dont think that is that important for this discussion so i'll skip it and get back to the main issue

Yes. What I mean here is that "good enough" implies that the information relevant for control system design is retained by the linear model. I agree that this can be very accurate when applied correctly. However, some information is lost by such a model. Particularly, the ripple components are removed. The ripple is typically at a frequency beyond what the control system needs to deal with, so the linear model is good enough (and very accurate) for that purpose, since no critical information has been removed. But, ripple is important in converter designs for reasons that have nothing to do with the feedback optimization, so the averaged model is generally not sufficient for all design issues. That's the basic thing I was trying to say, too succinctly.

 

[MrAl]

Hi again,

Ok very good

I'd like to turn now to another circuit example.

We start with a transistor, NPN type, lower power say less than 1 amp collector current rating like a common 2N4401 or really even the common 2N2222 or similar.

This device is non linear. In fact, when we look up the definition of non linear we should see a picture of this device

Now we connect the base through a resistor R1 say 100 ohms to the output of an op amp. We connect the collector to Vcc. We connect the emitter to a resistor R2 say 100 ohms, and the other end of the resistor goes to ground (R2 is the 'load'). We then connect the op amp non inverting terminal to a stable +5v reference, we connect the inverting terminal to the emitter of the transistor. (Heck i couldnt resistor drawing it, see attachment).

This circuit is a linear voltage regulator. We started with non linear components and ended with something we usually call linear. Is it really linear? Probably not because if the power supply varies from say 7v to 20v we get an output that is +5v regardless. But we usually call it linear anyway to distinguish it from a switcher. So we have:

Linear vs Non Linear, and
Linear vs Switcher

at the very least.

 

[PG1995]

Thank you, MrAl, Steve.

Now I would like to clear only one last thing about the circuit in Fig. 1 just out of curiosity. Steve has advised me not to consider it a linear system even if it looks like one because it can get me into trouble besides creating communication problems for me. But I would like to know what specific issue(s) which can crop up at some point if I start considering that circuit a linear system without worrying about anything and forgetting the fact that it's not a linear system in view of the definitions of the linearity. Thank you.

Regards
PG

 

[steveB]

The issue is you can get the wrong answer by applying linear system theory to a system that is not a linear system. Linear system theory begins with the concept of superposition. So, superposition will not work in Fig. 1., if you try to apply it. We could also talk about other linear analysis tools too, but why go any further? All linear analysis tools rely on the principle of superposition. Superposition is a key concept we use to simplify problem solving, and blindly applying superposition to a system that you have not verified is a linear system will give you the wrong answers if that system turns out to be nonlinear.

Take the system Vo=g Vi + k, and let Vi= sin(t). Here we see Vo=g sin(t) + k. Now let Vi=t. Here we see Vo=g t + k.

Now, someone asks you what the output will be when Vi=sin(t) + t. You decide to use superposition because you believe the circuit is linear. So, you say the output will be Vo=g sin(t) + g t + 2k. But, the correct answer is Vo = g sin(t) +g t + k.

Now, someone asks you what the output will be if Vi=A sin(t). Rather than solving directly, you decide to use scaling because the system is (so you think) linear. Your answer will be Vo=g A sin(t) + A k, but the correct answer is Vo=g A sin(t) + k.

 

[steveB]

... chimps can never be as smart as humans no matter how much you teach and train them!), you should be proud of it because then as a mentor you can be sure that your hard work, guidance and time didn't go down the drain, and you were able to teach a chimp where many had failed!

**broken link removed**

PG,

Just remember that humans and chimps share 98.8% of the same DNA. It's what we choose to do with that 1.2% difference that makes us better or worse than the chimp.

 

[MrAl]

Hello again Steve,

So by that logic i guess that means the US government does not know how to choose what it does with that 1.2 percent as shown by their recent major inactivities

Hey i ran into another linear vs non linear thing. When the op amp is in it's linear mode that means the feedback is satisfied. When it is open loop it's non linear. Interesting view i think here. So during a transition this could mean the op amp is in open loop even though normally in the linear mode. So here we're calling anything that satisfies the feedback as linear which includes almost every circuit an op amp is in unless it is open loop like for example used as a comparator.

 

[steveB]

Hahaha MrAl,

I agree, but I would have put a sad face at the end of the sentence.

 

[MrAl]

Hi Steve,

Yeah you're right. BTW i also added some to that post with some serious stuff :-(

 

[steveB]

BTW i also added some to that post with some serious stuff :-(

Yes, terminology can be maddening anyway. Think about MOSFETs and BJTs. The MOSFET has it's linear region in the ohmic region, which makes sense because it is "resistor-like", and then in the range where we say transistors are operating "linearly" the MOSFET is said to be in saturation. Yet, the BJT is in its linear region when operated in circuits "linearly", and saturation is when the circuit is being switched fully on.

Both devices can operate in nearly identical circuit topologies and perform similar functions, yet the terminology is opposite.

But, it all makes sense when you know the details.

 

[MrAl]

Hi Steve,

Yeah that's why i like to tell people that the interpretation depends a lot on the context.

Another interesting circuit here is the op amp based precision rectifier.

 

[steveB]

Hi Steve,

Yeah that's why i like to tell people that the interpretation depends a lot on the context.

I hope i didn't say anything that would contradict that, because i would say the same thing.

 

[MrAl]

Hi Steve,

Oh ok that's cool. I thought at one point you were trying to tell PG that there was no room for interpretation but not sure now

I've seen so many interpretations of "linear" that i just had to comment.

 

[steveB]

Hi Steve,

Oh ok that's cool. I thought at one point you were trying to tell PG that there was no room for interpretation but not sure now

I've seen so many interpretations of "linear" that i just had to comment.

No, I'm not telling him there is no room for interpretation for "linear". What I'm doing is responding to his questions relative to the context he is providing us. I very carefully read his questions to determine the context he is asking within. Most of his context relates to linear systems with superposition. Particularly his Fig. 1 where he says "linear system", and not "linear device" or just "linear". That context is clear to me, especially when taken with his originating post that references superposition. In another post he talks about "linear device" talking about OPAMPs, and there I agreed that the OPAMP can be considered a linear device under the assumptions he mentioned (within the rails and large feedback gain and lower frequency).

As I mentioned above, I don't have issues with your wording. You rarely say "linear system" and just say "linear", so I can read your context and know what you mean. Clearly "linear" has many possible contexts, but "linear system" should have fewer contexts for electrical engineers, and I would recommend that term be limited to systems that obey superposition, or at least provide very clear explanations to not mislead others you are communicating with.

 

[MrAl]

Hi again Steve and PG,

Ok great. I agree that certain wording leads to more exacting interpretations.
So you pointed out the need for more exact interpretations for certain problems and i pointed out the need for more flexible interpretations overall. So i think we both contributed the basic ideas of how to deal with the word 'linear' when applied to electrical circuits and systems. So i give us both an A+

I guess the real question here now is did PG get the understanding he was looking for after this discussion?

How about it PG?

 

[PG1995]

How about it PG?

Yes, I understand it well now. Thanks.

Regards
PG