If the sentiment expressed in the thread title were to be expressed by someone with two degrees, one in electronics and the other in mathematics, someone who holds a position of authority in the training of electrical engineering newcomers, what would such a statement say about the state of engineering education today? (Especially when that statement had been made in response to the identity, "cos(wt) = 1/2 * ( e^(jwt) + e^(-jwt) )2 such that the context was clear?)
Clearly, the quantity 'e^(-jwt)' does in fact decrease as 't' moves in the positive direction (i.e. increases) if 'w' is constant and 't' and 'w' are limited to the domain of real numbers (not imaginary). To suggest otherwise is foolish. To read more into the quantity 'e^(-jwt)' than is there, is equally foolish.
To try and make some philosophical attack about the education of someone based on that statement, only shows that you seem to think it does not 'decrease with increasing t'. Obviously you have an interpretation of mathematics and English that is in the minority.
The context of where the expression is used is immaterial. Notice that a corollary is also true, the term 'e^(-jwt)' increases with decreasing 't' (again, if 'w' is constant and 't' and 'w' are limited to the domain of real numbers).
The term could be in *any* formula, and the statement that the term 'e^(-jwt) decreases with increasing t' (provided 'w' is constant and 't' and 'w' are limited to the real domain) would still be the only rational position to take.
daestrom P.S. And I don't know of any application where the frequency ('w') or time ('t') are *not* limited to the real domain. :-)
Even with 't' and'w' limited to the domain of real numbers, 'j' is complex.
If you had also intimated that 'j' was limited to the said domain, then your statement below would be correct, but would then lack any credence in a NG devote to electrical engineering.
I guess I'm having a case of the terminal stupids: I seem to remember (and Perry's Engineering Manual, 2nd Edition, (c) 1967 confirms this)
that
e ^(-yi) = cos(y) - (i) sin(y).
so its value hardly decreases for increasing real y
Now anyone can change conventions and redefine terms, but within the profession this newsgroup represents one had better constrain himself to using conventional terminology if clear communication is the goal.
In many years of working in DSP I must have seen this 100s of times. Read the earlier threads see if you can spot Airy's error, it is almost as obvious as it is common. I'd look at the rotating vector model very carefully, if I were you.
in article x6oEb.15565$ snipped-for-privacy@newsfep2-win.server.ntli.net, Chimera at snipped-for-privacy@hotmail.com wrote on 12/18/03 12:35 PM:
One way such a problem i handled when it turns up, is to assume that the term has physical meaning. To do that a small damping term is added to get a term like exp(a-jwt) for ranges of zero to infinity. Then integrals such as averages can be computed that converge. Then limits can be taken as a->0.
There are lots of mathematical tricks like that around.
With all due respect, if a person is going to add a damping term let it appear in the expression.
If one is asking about the limit of e^(-at)*e^(-ibt) state the question that way.
This is a real life observation. Although EEs who bend conventions and the 'rules' are sometimes -- but not as often as they think -- very good and very creative, just a little of that goes a long way towards less than good performance reviews. Said differently, when it comes time for a technical manager to decide where his payroll bucks are being spent most effectively, guys or gals have to have lots of positives going on to counterbalance some of these kinds of things.
in article snipped-for-privacy@mb-m28.wmconnect.com, tony at snipped-for-privacy@wmconnect.comremoove wrote on 12/19/03 2:12 AM:
With due respect, life is not handed you in well designed exam questions. Many theoretical calculations come up with nonsensical answers because something has been neglected. Making sense out of it may be the hardest part of the problem.
As an example, consider charging a capacitor from a low internal impedance battery of voltage V by connecting the capacitor across the battery. If a charge CV gets transferred, the energy delivered by the battery is C*V. The energy stored by the capacitor is C*V^2/2 What happened to the rest of the energy?
This is trivial stuff. If you were showing me as either my student or my employee your analysis I'd point out your calculation suggests somehow you were able to impose an instantantionous change of voltage on the cap. Then we'd have a discussion about power contained in the vanishingly small inductances, radiative losses because the resonant frequencies are rather high, and a further discussion about bad math models.
As I said earlier in this thread, one should not say terms like sin(x) or jcos(x) tend towards zero at large x because it isn't true and the claim is indefensible. Add a damping term if you want to bring the model closer to your reality.
in article snipped-for-privacy@mb-m14.wmconnect.com, tony at snipped-for-privacy@wmconnect.comremoove wrote on 12/20/03 2:11 AM:
There is nothing wrong with the model. It is just incomplete. Errors of the nature indicated happen all the time. Certain answers to transform problems depend upon how you select integration contours. Miniscule damping often indicates which side of a pole the integration path must take.
With respect to your example, Show me one power source that puts out an infinitely lone sine-wave waveform. We have never had one like that. The recent power failures are indicative that we will probably never have one.
Well Bill, if you write an equation then describe in words the function, it would be wise to describe the function as written, or modify it to express your reality.
Somehow I haven't communicated that fairly simple notion to you. Since I can't teach it, or you refuse to learn it, and I surely will never agree sin(x) as written somehow approaches zero for large x, let's kill file each other. Consider it a mutual failing grade.
in article snipped-for-privacy@mb-m04.wmconnect.com, tony at snipped-for-privacy@wmconnect.comremoove wrote on 12/20/03 2:16 PM:
You communicated well enough. You do not realize, however, that making sense out of improper integrals is a cottage industry with mathematicians. At least applied mathematicians.
Another example of using practical physical understanding to make sense of an impossible mathematical artifact is the Gibbs phenomenon, As you probably know, representing a square wave as a sum of harmonic give rise to sharp ringing at the transition points. Increasing the number of coefficients makes this ringing narrower, but it doe nothing to reduce its peak. In any physical situation, such spikes are not present. Feed the sum of harmonics into any reasonable filter, and the Gibbs phenomenon disappears.
in article snipped-for-privacy@mb-m04.wmconnect.com, tony at snipped-for-privacy@wmconnect.comremoove wrote on 12/20/03 2:16 PM:
Somehow, against my better judgement, I am getting entrapped by this subject. I will present a rigorous calculation of the integral from 0 to infinity of sin(wt). That along with the integral of cos(wt) allows the integration of exp(-jwt). I will leave the cosine integral for the reader.
Assume I have a box that implements the integration. On the box there is a knob that adjusts the damping factor s that allows for the infinte range of integration. When I am done, I turn the knob to reducing the damping factor s to zero. Such a box, presuming it can be realized is what separates a mathematical calculation from a physical reality. Thus, I am looking for the integral of exp(-st) sin(wt) over an infinit range.
You should be able to understand that this integral is merely the laplace transform of sin(wt). Doing the integral or looking it up in a table of laplace transforms, the value of the integral is w/(s^2 + w^2). I now take my box and, by using my knob, turn s down to zero. The value of the integral is 1/w.
How do you interpret that? It makes no sense mathematically but a lot of sense physically.
The arguments you've presented, from suggesting instantaneous voltage changes across caps to your box with knobs, has anything to do with the assertion that e(-jx) approaches zero for large x. The function has nothing to do with physical reliability, power failures, or the like. It is a simple mathematical function.
Early on in my remarks about this subject I wrote about engineers who used nonconventional analytical and synthesizing methodologies. It took powerful offsetting evidence for me to not reject them out of hand. Nothing in this thread has provided evidence to suggest I should change my mind.
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