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"Veritasium" (YT) - "The Big Misconception About Electricity" ?
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TimFox:
The simple complex algebra that I do on my I and Q values is easier than using trigonometry on the sine and cosine waveforms, which is also valid mathematics.
What you see on an oscilloscope is a function of time.
Often, as in frequency response or impedance calculations, what you want is a function of frequency, where complex algebra is useful.
adx:

--- Quote from: TimFox on April 08, 2022, 04:46:49 pm ---The simple complex algebra that I do on my I and Q values is easier than using trigonometry on the sine and cosine waveforms, which is also valid mathematics.
What you see on an oscilloscope is a function of time.
Often, as in frequency response or impedance calculations, what you want is a function of frequency, where complex algebra is useful.

--- End quote ---

Yes, tricky - "valid mathematics" I think being the sticking point (or question) for me.

Time domain, everything is real-valued, and therefore "real" in my mind.

Frequency domain is an abstraction which is much further removed from the direct analogue "this point 'has' this quantity of voltage right now" (itself a short abstraction away from potential difference of an electric field). Decomposing an arbitrary signal into frequency components is a transform which has no physical significance whatsoever (in the sense that inventing a fairy tale to describe some physical phenomenon is no more valid than some other description which works - a point made by penfold a while back). I like to keep this fact (it's a fairy tale) in mind where possible. Do I believe it? Yes - it takes one set of real quantities and converts it into another, isomorphically. The fairy tale gains physical meaning when we lose the arbitrariness of the signal and begin dealing with sinusoids - RF, sound, bandlimiting, synchronous demodulation... I am happy to think in terms of reactance and the give and take of energy - not a complex number in sight. But I still like to check myself when talking about "frequencies" so as not to get too carried away by the fairy tale.

Complex numbers are where my belief in a fairy tale ends. Not because they don't work, but because the mathematical validity seems to be based on a leap of faith - a circular definition (no pun intended). Not entirely, but not 100% convincing. The fact that most 'proof' seems to consist of wildly gesticulating at my paragraph above saying "but but phasors" suggests that the proponents of the fairy tale have become so under its spell that they have lost the ability to reason.
hamster_nz:

--- Quote from: adx on April 09, 2022, 03:40:14 am ---Time domain, everything is real-valued, and therefore "real" in my mind.

--- End quote ---

I'm sure you are aware of the Hilbert Transform.

https://en.wikipedia.org/wiki/Hilbert_transform

It's the mathematical equivalent of what you brain does when viewing the Spinning Ballerina Optical illusion.



You can use it to split the real-valued data into their positive and negative frequencies.

These frequencies do exist, as seen in up-converted baseband signals. With the Hilbert Transform you can then throw one side of the spectrum away, allowing you to generate just upper side band or just lower side band signals.
adx:

--- Quote from: SandyCox on April 05, 2022, 01:58:43 pm ---This discussion is really more about mathematical philosophy than engineering. For engineers, the subject of mathematical philosophy doesn't put bread on the table.

Here's another interesting one:

Lets say that we have a rectangle with width w and height h. Does its area remain the same if we turn the rectangle through 90 degrees? If it does, then we have "proven" commutativity for the real numbers, i.e. w*h = h*w.

I tend not to worry too much about these type of questions and rather focus on making my designs work. I need to be able to factor polynomials over the complex numbers to design filters and control loops. I need the poles and zeroes of transfer functions. I need the Fourier transform for the frequency domain perspective. I need theorems from Complex analysis. So I tend not to worry about the philosophical meaning of j. It works and that's good enough for me.

--- End quote ---

True. Your mathematical skills are well above mine, I use those tools more at arm's length and not every day, and even more rarely analytically. TimFox posted a circuit at the bottom of p67 that looks like a SMPS compensation network - anything more than that I tend to use a custom tool a manufacturer might make available, simulator, or solve numerically (using some of those 2 DP GFLOPs on a 12 year old CPU). Or just select on test. But I always hope to understand the concepts of what I'm doing, and that's put in context when dealing with a non-engineering-trained client who has cobbled together something that works but has no concept of engineering theory.
adx:

--- Quote from: TimFox on April 06, 2022, 08:51:31 pm ---"Twice I provided (well you did for one) examples of the concept not being used or necessary."

Yes, also the concept of Bessel functions is not needed when I balance my checkbook, but they are quite useful in calculating the harmonic content of a transistor collector current (driven by a small sinusoidal base-emitter voltage).
Yet another piece of 19th century mathematics that is irrelevant to one or even three applications. 
However, the basic concepts of logic disagree with your argument here, and they continue to be relevant to digital design.
...

--- End quote ---

Thought I better tackle this directly. That's not what I meant. I was exactly on point when proving the assertion I made. Perhaps an example of where the voice of reason is not always the same as the voice of logic - HuronKing's position sounds reasonable especially in light of my belligerent claims, but I was logically correct. Twice.

Searching for the Backfire Effect: Measurement and Design Considerations
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7462781/
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