# Proving the linearity of a resistor

Discussion in 'The Projects Forum' started by mapisto, Apr 21, 2011.

1. ### mapisto Thread Starter Member

Dec 25, 2009
36
0
Hi,

How could i prove the linearity of a resistor?

I've tried using the 2 laws of linearity:

1)superposition
2)homogeneousness

But weirdly, it works for everything, even for sine! which of course isn't linear!

what am i doing wrong? is there any other way to prove linearity?

Please guide a lost student

thanks!!!!

2. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
A resistor's linearity is not dependent on the driving function's linearity, rather on the relationship between the response (e.g. the current) & the driving function (e.g. the applied voltage).

3. ### mapisto Thread Starter Member

Dec 25, 2009
36
0
Ok, thanks, but how can i prove the linearity ?

4. ### THE_RB AAC Fanatic!

Feb 11, 2008
5,435
1,305
Connect it to a range of different voltages, and measure its resistance at each voltage (by measuring current), then plot the result to see if you get a linear plot. This is a really basic test.

5. ### Adjuster Well-Known Member

Dec 26, 2010
2,147
300
This amounts to a demonstration of Ohm's Law. It is so fundamental to understanding electronics that effort spent in getting to grips with it is not wasted. Without a good grasp of the basics, it is difficult to make much progress.

http://en.wikipedia.org/wiki/Ohm's_Law

6. ### CDRIVE Senior Member

Jul 1, 2008
2,223
99
Do you have a variable voltage power supply at your disposal?

7. ### someonesdad Senior Member

Jul 7, 2009
1,585
141
It's a good question and deserves a little discussion. First, you can't "prove" things about the world -- you have to measure them experimentally. With care, you and others find that things seem to behave rather consistently, so you propose a rule about that behavior. If enough people find experimental data consistent with it for long enough, it then slowly gets promoted to a "law". But no "law" is ever proven.

Now, let's look at the cherished "Ohm's Law". Ohm's experiments (around 1840 IIRC) demonstrated with the technology of the time that the current through a metallic conductor was linearly proportional to the voltage across the conductor. Of course, we know this isn't really true. If you don't believe me, just keep increasing the voltage until you do. Thus, it's an approximation. One of the most common examples that it's not a law of nature is the common incandescent light bulb. Another example where you'll make errors if you think it's a law of nature is making temperature measurements with RTDs.

What usually is quoted as "Ohm's Law" is actually the definition of resistance. That is, the resistance of a conductor is the voltage across it divided by the current passing through it. This applies whether the relationship between current and voltage is linear or not.

Also, remember this definition of resistance is only a useful macroscopic approximation. As you start to go down in size and in time, you start to find variations from the law due to things like Johnson noise, shot noise, etc.

So, does this mean it's all a bunch of hooey? NO!! I think the definition of resistance (OK, "Ohm's Law" for you EEs) is just about one of the most useful equations ever. Get to know it intimately and it will serve you well your whole life -- but also be aware of its limitations.

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8. ### mapisto Thread Starter Member

Dec 25, 2009
36
0
Thank you all for answering and i realy appriciate it.

All you've said about testing it on individual numbers won't prove anything (unfortunately : \ ).

but i think i've found a mathematical way for proving it, i just need your confirmation..

From linear algebra, in order to confirm something is linear, there are 2 conditions:
1)superposition.
2)homogenious.

now, we're talking about 2 dimentions here, (I,V), so if i can prove the 2 conditions always work for I and V... did i prove it? : \

on the other hand, it's kinda weird, coz the only way to deny it, will be to find an example which shows it doesn't always work, and i can't realy find one.. can u?

so if I give it to my doctor as an answer.. (it's a part of a project..), could he deny it? or that's the right way of doing it?

Thanks!

9. ### CDRIVE Senior Member

Jul 1, 2008
2,223
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Mapisto, don't get lost in some of the unnecessary smoke that was thrown up. It will only serve to obscure and confuse the issue.

• ###### EIRcurves.JPG
File size:
48.6 KB
Views:
44
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10. ### steveb Senior Member

Jul 3, 2008
2,433
469
You are being much too vague. You haven't been specific on exactly what you are trying to do, or what you have done already. Exactly what are you trying to prove is linear about a resistor?

-A real resistor response via experimental data?
-Ohms Law itself in the mathematical sense?
-A hypothetical resistance function which does not obey Ohms Law?
-A derivation of Ohm's Law via application of basic physics principles?

When you want to prove something, it's important to make a clear statement of exactly what you are trying to prove. Saying you want to "prove the linearity of a resistor" is a vague statement, and hence is unprovable.

Once we know exactly what you are trying to prove, we can identify the tools used to prove it. Ohm's law can be proved to be a linear law mathematically. The validity of Ohms law in the approximate sense that it matches typical measurements can be proved from an experimental point of view. Clearly there is a difference in meaning of the word "proof" in these two cases. The first is concerned with a rigorous logic proof in the absolute sense, while the second calls for an appeal to preponderance of the evidence and the application of common sense about range of circumstances for validity.

11. ### CDRIVE Senior Member

Jul 1, 2008
2,223
99
My goodness, I just realized that you're question (specifically) wasn't answered! You requested an algebraic method of proving the theory without the use of instruments. You can reproduce the spice image that I posted by drawing your own graph. Use Ohms Law to calculate each point on the graph. When you're finished you should be able to draw a straight line through the points of the graph. Is that what you wanted?

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12. ### mapisto Thread Starter Member

Dec 25, 2009
36
0
Guys, first i wanted to thank you again for helping me so much. I'm so glad I've found this forum.
I've just started my engineering studies and i'm quite new to the area...

What my Phd asked me to find is (i'll quote) "prove the linearity for elcetric resistance, capacitance and Inductance".

I believe he meant that he wants us to show him that the relation between the current and the voltage upon a given resistor will always be the same, no matter how I change each of them.

While the concept is pretty clear when you think about it and looking at a graph, i still believe that's not what he's asking. he's asking to PROVE and not SHOW that it happens.
I don't believe he wants us to use an induction (god help us if he does), I think, as i've writen in my last post, that the method he wants us to use (because he talked about it..) is the 2 conditions of superposition and homogenious.

I wish I could just go on to matlab and show him a graph which is linear
: \ but he won't accept it as a proof.

What do you think?

13. ### CDRIVE Senior Member

Jul 1, 2008
2,223
99
I think your prof thinks his students have as much spare time on their hands as he does. Let's hope that as your course progresses your prof teaches real world electronics. We need electronics engineers, not electronics philosophers.

14. ### steveb Senior Member

Jul 3, 2008
2,433
469
I think the only reasonable interpretation of this question is to prove mathematically that the normal simplified equations for resistance, capacitance and inductance are linear equations.

Personally, I think his question is vague and open to interpretation. However, only the above interpretation allows you to prove linearity, so the context of the question is what clarifies the intent.

So you need to prove linearity of the following relations.

V=IR
I=C dV/dt or Q=CV
V=L dI/dt or F=LI

V-voltage
I-current
Q-charge
F-flux

R, L and C are constants

You can prove linearity in a formal way. Eventually, you will recognize linearity by inspection. Note that any circuit combination of these linear components will result in a system which is linear.

Linear systems are defined as those with the property as follows.

F(a*g(t)+b*h(t))=a*F(g(t))+b*F(h(t))

where a and b are constants, g and h are time dependent input functions and F is the operation that the system performs on the inputs to generate outputs.

So take the resistor case.

i(t)=a*g(t)+b*h(t)

Now apply the resistance "operator" to get the output voltage.

v(t)=R*(a*g(t)+b*h(t))

Now use simple algebra to rearrange to the following.

v(t)=a*R*g(t)+b*R*h(t)

It's clear that this proves linearity based on meeting the definition.

Now ask if linearity would hold if resistance had an offset. This is expressed mathematically as

v(t)=R*i(t)+K where K is a constant.

You might think this is also linear because this is an equation for a line, but you would be wrong. Let's prove it.

v(t)=R*(a*g+b*h)+K=a*R*g+b*R*h+K

however linearity would require v(t)=a*(R*g+K)+b*(R*h+K)

but this leads to v(t)=a*R*g+a*K+b*R*h+b*K

It should be clear that this does not agree unless K is zero.

Note, you can apply the same linearity definition to integrals and derivatives. Hence, they are also linear operators. Thus, ideal coils and ideal capacitors are also linear. Try it out to prove it formally.

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15. ### mapisto Thread Starter Member

Dec 25, 2009
36
0

lol.. i wish i could show him that without making myself his most hated student of all times

16. ### mapisto Thread Starter Member

Dec 25, 2009
36
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Thanks! i'll try proving it myself using ur explanation.

I hope soon enough I'll be able to answer questions here myself and pay you back for ur time and patience.

Mapisto.