Complex Numbers, Phasors, Impedances, and Frequency Response
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- Опубліковано 6 кві 2024
- This is Episode 6 in the ECE Topics series. It covers complex numbers, phasors, and impedances - the necessary mathematical machinery for solving any "AC circuit". In keeping with other episodes on this channel, the goal is to understand circuits, and several practical examples are used as examples. While the math is covered rigorously, the pre-requisites needed are limited to algebra, and basic trig.
- Наука та технологія
Thanks again for another excellent video. Very brave of you to try and teach "doing the math." Complex numbers/analysis are just a tool and the ideas needed for circuit analysis are really not that difficult.
Something that has helped me understand the beauty of using complex analysis is that for a differential equation, let's take v = L*(di/dt) for example, if we assume any of our input signals is a complex exponential function: i(t) = exp(j*omega*t) then v(t) = L * j * omega * exp(j * omega * t). If we replace exp(j * omega * t) with i(t) we get v(t) = [ j * omega * L ] * i(t). Nearly magically, it's clear that the quantity [j * omega * L] is just like a resistance in that it relates voltage and current through an element.
Once you realize that differentiation is linear (the derivative of a sum is the sum of derivatives) you can sum any number (countably many = Fourier Analysis, uncountably many = Fourier Transform) of exponential inputs together as a single input and know the output response.
Mathematicians (of which I consider myself semi) would make it sound way to intellectual and tell you that complex exponentials are the Eigen functions of the differentiation operator (remember that for your next cocktail party) but this all kinda works because the derivative of the complex exponential is a complex number times that complex exponential. That's the root of it mathematically.
Thanks for adding some excellent thoughts here ! I totally agree. Going the complex sinusoid route in the derivation would be equally valid - and perhaps even better. I was just worried that it might look too much like a foreign language to many viewers. It's always hard to know who is in the audience and what their background in math is. As you said, though - once you get comfortable with seeing e^jwt stuff, the impedances just kinda fall out magically :-) And like you said - they're needed for Fourier transforms, and are a much more elegant expression for Fourier series as well.
Your content makes me sad. i’m 35 with kids. A software engineer for life, I got into RF during pandemic. My grandpa was into radios but never let me touch them, triggered me into the hobby late in life. I’ve read and watched everything, spent a mortgage on a lab…
4yrs in and you’re hands down the best teacher and have the best style I’ve come across. So much so that it hurts to know my young kids might never experience this. I worry AI, in its pursuit of efficiency will atrophy our ability to be curious , and the reward you get answering or teaching something on your own
You might be the last of a dying breed. One of the final inputs to the human corpus before it gets saved as V1 and forever iterated on by AI going forward; while humans sit idly. God speed brother
Thanks for the comments. Oh gosh. I hope we're not the last of a dying breed. But I have been wondering similar things about AI's effects on humanity if/when we get to AGI. I was watching an interview and Q&A with Sam Altman last night on UA-cam (from an ETL seminar held at Stanford). The issue you raised was not addressed - until one person at the end asked about it. Altman's answer didn't really help IMO. He seems blind to that. But some of his earlier comments on slow rollouts and viewing AI as another tech to help in joint discovery were slightly encouraging. Glad to hear you've got a lab and are getting into RF technology ! Thanks for your thoughtful remarks. You have a good way of expressing these issues. 73's
Backing off a bit from the "without estimation", if I know the form of the response, and have the mag/phase over a frequency range, is there some magic Python code somewhere that would curve fit to the known response? ie- knowing the RCL (s) and topology the form of the Laplace f(s) and the response should (?) be enough to fit?
[ I am a retired EE, and my calculus skills have dissipated over the years,.... ]
This would make a good video topic, ;-} for me at least.
I am working on several projects and ideas for a UA-cam channel of my own.
One if which is the analysis and restoration of audio gear. (vacuum tube)
I have some McIntosh pieces to restore. I have a MC20 preamp which is notable with several settings for equalization (RIAA etc) on the phono input. I have Spice simulated the McIntosh design and note that it is not quite correct. I want to run sweeps with an analyzer to get mag/phase fit that to an f(s) and compare to what the published time constants are for the various equalization that have been used on LP's over the years.
No this is not an absolutely necessary approach, but it could be instructive or interesting to viewers. I hope to drive the Audio-Fools nuts,... LOL
Cheers, dan
Projects sound great ! Technically, if the poles and zeros in the response are far enough apart, you might be able to estimate RC products, and then simulate and compare with measurements to refine them perhaps. Sorry - I don't know of any Python scripts, etc. But maybe someone else reading this discussion will chime in with one :-) Here's one example that I thought of from the ECE Topics videos. Don't know if it will help, but if this is put together with the complex math/graphing stuff, then maybe it could be useful... ua-cam.com/video/8oGsBL6CHk4/v-deo.html
@@MegawattKS thx I will check it out
question: are you aware of a tool or tools which would provide Laplace equations from a Bode Mag/Phase plot? (without all the manual estimation)
Sorry - no. FWIW, I once labored through a textbook on Network Synthesis. Doing a search for that combined with "bode", I found this, which might be useful background. I think they cover the issue of non 1:1 results. And be warned that there are often (always?) at least two possible s-domain results from the fitting. One is stable and the other is unstable. Bode plots don't typically provide enough information to determine if the system is actually stable (all poles in LHP). Hope that helps, at least a little ... en.wikipedia.org/wiki/Network_synthesis
@@MegawattKS thx