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m.s.j · Joined: 28 Jun 2007 Posts: 9

This problem is subjected to precommisioning test of an instrumentation system in hypothetical industrial plant. Replacing of electrical parameter with other physical parameters is possible, so in addition electrical engineers everyone who involved linear systems can think about this problem and share their knowledge and experience.
The instrument device which shall be checked, includes one current input (0-60 mA ) and one indication pointer that moves proportional to input currents. Purpose of a.m. mentioned test is investigation of linear characteristics of that equipment functions. Random current inputs at many different temperatures are applied to device and following report is recorded by it.

item,,,,, input current (mA) ,,,,,,,pointer replacing (cm),,,,,,,, temperature (c.d.)
1 ,,,,,,,,,,,,,,,,,7 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,5 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,3
2 ,,,,,,,,,,,,,,,,14,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 10,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 6
3 ,,,,,,,,,,,,,,,,21 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,15,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 9
4,,,,,,,,,,,,,,,, 28 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,20 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,12
5 ,,,,,,,,,,,,,,,,42 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,30 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,18
6 ,,,,,,,,,,,,,,,,49,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 35 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,21

Contractor representative affirms according to above test result, function of device is linear. But representative of client says " those results only prove one of two properties of linear system and superposition property shall be proved too".

Contractor representative believes it is not necessary and " f(ax)=af(x) " is enough to prove systems linear properties, because we can write:

f (ax) = a f (x) eq. no.1

and a=b+c

f [(b+c)x] = (b+c) f (x) eq. no.2

also f [(bx)+(cx)] = b f (x) + c f (x)

According to eq.no.1 we can write:

b f (x) = f (bx) , c f(x) = f (cx)

Therefore we can write:

f [(bx)+(cx)] = f (bx) + f (cx)

Because both of b & c quantities are individual terms, therefore (X1= bx) & ( X2=cx) are individual and we can write:

f ( X1+X2 ) = f (X1) + f (X2) eq. no.3

He continues :

The above equation intends superposition property of system.So we prove linear property of function device only with the aid of equation no.1.

Whether the opinion of contractor representative is right?

gabrielpacyna · Posted: 2007-10-15, 22:07

I am deeply convinced that the system is linear.
I only may say that according to many electrotechnical books the only and sufficient proof that any element is linear need to be:

the proportional rise of the signal given to input is causing the proportional rise of the signal output.

If about the superposition, this [b]is determinated by the definition of linear system[/b] and You can find it in every mathematical books, especially in Algebry theory. But actually I am not sure, in this. But contractor prooved it I think correctly.

I am very sorry for errors, especially in such thema like this, but tried write as correct as I could.

[ Added: 2007-10-17, 23:42 ]
Actually I must say that for the system is linear it is needed that :

1. f(ax)=af(x) and 2. f(x+y)=f(x)+f(y).
These two must be done. In fact, the client is right in the matter.
But you have written that you had only one input and in fact the second principle did no matter in fact because you had only signal x, you did not take any more signals.
So the first principle we are to take for our thinking.

In opposite there are some devices like summary devices, multiply devices where there are two or more inputs and we have to proove the second, but in our case we do not. I think this is linear system.
You should ask at least any mathematics or specialist of automatics if they agree with that opinion.

I summary you can say that

1. f(ax)=af(x)
2.f(x+y)=f(x+0)=f(x) + f(0) may be the f(0) should be equal 0 ??? but x+0 in your case is concerning the some input so it is f(x+0)=f(x) (we have no other signals like x).

gabrielpacyna · Posted: 2007-11-28, 00:50

Do you think, that x, y, z electric connections are sexy. Do not make sex with phase voltages. You are not insulator. :wink:

admin · Posted: 2007-11-29, 13:27

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