Saturday, June 10, 2006

Inconsistent Circuits Formula

Last week, while studying for my circuits exams i stumbled upon one of the least well-defined physics formula that I had to study... The formula goes like this:
The effective value of a current which is composed of any number of sinusoidal currents of different frequencies can be expressed as:
Ieff = SQR(I1eff2 + I2eff2 + ... + INeff2)

source: Engineering Circuit Analysis (6th Edition), by William H. Hayt / AC Circuit Power Analysis - Page 369

This formula has caused me a lot of headaches because of the (illogical?) word: "different".... Lets consider two sinusoidal currents: I1=10sqr(2)cos(wt), I2=10sqr(2)cos(5t); So, I1eff=10, I2eff=10...
Now according to the way it is calculated:
Ieff = 10sqr(2), for all |w| != 5
Ieff = 20, for all |w| == 5
In other words:
Ieff(w=5+) = Ieff(w=5-) = 10sqr(2)
while, Ieff(w=5) = 20

The discontinuity in the function of Ieff is very obvious! The existance of this discontinuity is very questionable, especially in the current context.. Thats unnatural! You cannot prove two physical (even mathmatical) quantities to be equal! Since we cannot prove that two quantities are equal, it makes sense that an equation that works for different values of some variables, work also for equal ones....
In short, I think that the given equation is missing: Maybe, the details were skipped for simplicity, or maybe that author himself doesnt know what the fuck he's talking... If so, this might be an interesting area to investigate!

Update: (Jan 10, 2007) This problem has been resolved. Read more here.

PS: sqr(x) is square root of x
PS: The scientific validity of claims is NOT asserted

12 comments:

Tala said...

i took circuits two years ago :)
same book, i gave it to a friend. what you are talking about is in chapter 10 or eleven?

its very vague, very short in explainations and not a lot of examples. take a good care of the practice questions. but the course is nice. if you took it with a good Dr. its a base couse. you have Dr Hamdan who is the brother of our Department's Dr Hamdan.. right? we all suffer!

practice as much as you can. this is how it is learnt.
Best of Luck

Devil's Mind said...

Thanks for the advice: Luckily, i already finished the course and did really well (i got 95/100)! Our teacher is called Jamal Thalji, not Hamdan or anything...

Anyways, this is all beyond the point of this entry, which is to suggest possible source of inconsistency in the current formulation of AC circuits and phasors...

Anonymous said...

"You cannot prove two physical (even mathmatical) quantities to be equal!
Since we cannot prove that two quantities are equal, it makes sense that an equation that works for different values of some variables, work also for equal ones...."

if u cant prove two quantities to be equal then our dictionaries should get rid of the words Equal, equations, etc because we dont need words to describe things that do not exist such as equality.

u contradict urself by assuming that there are two equal values of a variable... is 5=5 or did i get it wrong?
be careful with ur words.

with the same logic u cannot prove
that 10sqr(2) != 20 so the Formula might be continuous.

failing to prove that they are equal doesnot make them inequal..

try to take things as simple as possible...the formula works only with a set of of sinusoids of "different" frequencies.

have a nice day

Devil's Mind said...

"You cannot prove two physical (even mathmatical) quantities to be equal!"

I see you have a problem with this statement. Consider this:

Mathematical Justification:
What is the value of Zero/Zero ?!

The answer is "any value". It is NOT necessarily 1.

Since
Zero\Zero != 1,

then
Zero != Zero,

Add five to both sides:
5+Zero != 5+Zero
=> 5 != 5

Physical Justification:
Two cars are moving. We use a speedometer with adjustable accuracy to measure their speed.

Say the speedometer reads:
5.00 and 5.00

Does that mean they have equal speed?! NO.

Say we increased accuracy to 6 decimal points. The speedometer reads:
5.000000 and 5.000000

Does that mean they have equal speeds?! STILL no! In fact, no matter how much you increase accuracy you WILL NEVER reach a satisfying result that the two cars are moving at the same speed.

Anonymous said...

U r still contradicting urself by using equality and inequality in ur math, add to that, The Basis u r using is false...u cannot start from an undefined quantity such as 0/0 to calculate a defined quantity, this is the ABC of mathematics and algebra.

u also relied on 0*1=0, X+0=X, i would ask u to prove both, cannot use (Y-Y)*x=0, because u cannot prove that Y-Y=0,

remember u r the one who stated that u cannot prove mathemtical quantities to be equal...so the math u r using is not usable because it is all based on equality.

u always need axioms to level the playground and enable us to come with theorems..

"you WILL NEVER reach a satisfying result that the two cars are moving at the same speed."

that would be true if i were a mathmatician...but as an engineer i can use a limited number of digits(significant figures) and be happy with the approximation or estimation i have.

Devil's Mind said...

My mathematical justification is sound, but maybe it needs a little bit more detailed explanations. The physical justification is easier to understand.

Briefly, I want to remind you that the origins of mathematics come from calculus, especially the limits.

Every mathematical statement is at its roots a limits statement. So, something like:
8 = 8 , is actually
lim [x->8] x = 8...
we only write 8=8 for simplicity, but stricktly speaking 8!=8, except when considering the limit.

Now, regarding the physical justification... You can see that as an engineer you might accept certain degree of accuracy. But when applying that accuracy consideration to the circuits formula that I have mentioned, a small difference like 0.000000000000000001, can give you very much different results, which is counter-intuitive result!!

Devil's Mind said...

I will make a post about this tomorrow or after tomorrow, and in that post I will post exactly what I meant by the "we cannot prove two quantities to be equal".... And will give an account of the limits equations to consider to address the possible misunderstandings.

Nothing is really in contradiction, we just need more precise description of what it means to say two things are equal or unequal. We just need to take a few ventures in the meaning of "Zero", the value that is usually avoided in mathematics classrooms.

Please consider joining in this soon to come post.

Samir said...

Hello ..

I have two things to share:

- the equation is valid and good, even with your limits to approach w -> 5.

When the two sinusoids are of exactly equal frequency, they will add up and form a new sinusoid, which has the standard RMS value. (you calcualted 20)

When the frequencies are off, the two sinusoid will be adding up for when they are "in phase", and in time, the difference in frequency will shift the two signals apart, and they will start cancelling each other.. in effect, you have that equation to represent the effective value of the total current. (you calculated 10sqrt(2) )

So it's a matter of "time" for the two signals of different frequencies give you the collective RMS value. if you want to make the frequencies apart by 0+ , you need infinite time to have the total RMS effect. until infinitey, the collective signal will have a "wavy" RMS around the mean they should reach.. when the frequencies are far enough apart, you'll get the total RMS value "quickly". (taking the limit to zero, is un-practical, because it never happens in the lab)

The second thing. Devision by zero is undefined. it results in NaN (not a number) in computer science.

therefore, you cannot proceed from:

Zero\Zero != 1 , to:
Zero != Zero

because you are not dealing with numbers anymore. Plus, here you multiplied by zero on both sides, and you cancelled 0/0 to be 1 on the left side.. you just said that's invalid (in the first statement). Your proof contradicts itself.


I'd explain stuff better face to face, I wonder if you followed me or not.

Salam,
Samir.

Devil's Mind said...

"So it's a matter of "time" for the two signals of different frequencies give you the collective RMS value. if you want to make the frequencies apart by 0+ , you need infinite time to have the total RMS effect. until infinitey, the collective signal will have a "wavy" RMS around the mean they should reach.. when the frequencies are far enough apart, you'll get the total RMS value "quickly". (taking the limit to zero, is un-practical, because it never happens in the lab)"

Thank you very much!! You have said almost everything I had to say! I only yesterday completely understood what you said.

You explanation is what I found out only yesterday, but the problem is, neither the book nor our teacher explained those thing - I had to figure it out myself!!

Regarding the Zero\Zero issue, in order to clarify my point, consider the following:
(a) 0+ = 0-
(b) 0+ != 0- AND 0 = 0+
(c) 0+ != 0- AND 0 != 0+
Choose (a), (b), or (c). So which do you choose?! :)

Our confusion starts from different meanings of (a), (b), and (c), once we resolve this confusion, you will be able to see what I mean. For now, just tell me which one you find more reasonable.

Samir said...

You're very welcome.. you're on the right track, professors and books can only guide you, its up to you in the end to figure stuff out. A quote I read recently said, "don't let your schooling interfere with your education", by Mark Twain I think.

I'm sure you knew that, but I felt like sharing..

Dude, I have the devision by zero dilemma behind me and I have finals these days. I think you're getting into the issue in your next post.. So I'll be back on the 18th to see what you're up to ;)

So you're a second year CompE student?

Devil's Mind said...

Third year CompE... And yes, you are right, I will post about this issue soon.

Devil's Mind said...

A FORMAL PROOF of Zero!=Zero and explanation of what it means to say that can be found here.