Reactance Susceptance Impedance and Admittance Oh My! - Page 1
Forum section: electronics / beginners

author: edy
author_rank: Super Contributor
timestamp: February 16, 2021, 11:23:39 pm
content: I'm currently reading through a book titled "Teach Yourself Electricity and Electronics, Sixth Edition" by Stan Gibilisco and got to the chapters on Reactance (Capacitive/Inductive), Susceptance, Impedance, Admittance and the RX and GB half-planes and it's just blowing my mind!This is my first foray into AC and rigorous mathematical approach to electronics that goes beyond simple Ohms law and Resistance/Capacitance/Inductance in series/parallel circuit calculations. The chapters do their best to show the elegant symmetry between all these concepts, although I see why the reviews are a bit mixed with some say it is too difficult while others don't like the repetitiveness.At the moment I am just trying to remember and grasp the concepts, but can't help noting the "beauty" of how all the math just seems to line up and boil down to a simple equation in the form (2 x pi x f x A) or it's inverse, where A = inductance or capacitance depending on which form you are using and whether you are determining reactance or susceptance. Keeping track of the negative (-), when to use the equation or the inverse, etc... is tricky but all of that symmetry is interesting.I'm still not at a point where I actually know what to do with any of it. I'm also not having any intuition on "why" it is the way it is, and what 2 x pi has to do with it (e.g. why that particular constant) and why the complex plane seems to be a great place to track all of this. I understand working with the complex plane, measuring vector lengths, angles, finding the inverse of "j" or any complex number (as is needed to switch from complex admittance to complex impedence values)... But I fail to grasp any deeper intuition (which I assume to exist and some of you probably know) as to the reasoning that 2 x pi and the complex plane is the place that all this mathematics seems to work to figure this all out.My only other main experience with the complex plane is generating Mandelbrot fractals. I know that 2 x pi radians is a full circle or 360-degrees, or a complete sin/cos cycle. Phase angles are still a confusing mess, although I learned voltage leads current in inductors (I guess because the magnetic field delays the current movement, so it lags behind), and current leads voltage in capacitors (because electrons have to flow first before a voltage difference occurs between plates) but I'm not sure why it's important and what you would use it for.I'm hoping these and other ideas will make more sense as I keep reading, and hope to eventually have some intuitive sense of what is going on. Certain analogies can only go so far, and I don't want to start getting the wrong ideas in my head as these analogies eventually "break down" and no longer help but actually hinder understanding (this may actually be causing some of my confusion because I have to unlearn them).I would appreciate any help or suggestions at how to look at these concepts in a different way, especially to make sense of 2 x pi and the complex plane in general and I guess why working with it helps in so many applications. I was reading this page:https://en.wikipedia.org/wiki/Complex_numberand looking at the way it relates to trigonometric functions and natural exponents, phasor calculus (https://en.wikipedia.org/wiki/Phasor), and it's something I'm trying to wrap my head around!Fascinating stuff!

author: Benta
author_rank: Super Contributor
timestamp: February 16, 2021, 11:40:47 pm
content: It's all about mathematical modelling.The basic physical unit is time (not frequency), and solving circuit equations is possible using only time as the variable. And in the real world, the time domain is the actual one.But this results in having to use differential equations to describe the circuit behaviour, and differential equations are pure hell to solve.The mathematical transformation into the complex frequency domain is a smart way to avoid this, because you can then work with algebraic equations, which is immensely easier.This just for background information.I suggest that you do a search using the term "Laplace notation", which should give hits on working in the complex frequency domain.

author: Terry Bites
author_rank: Super Contributor
timestamp: February 17, 2021, 12:30:40 am
content: Dont get too bogged down in the maths- start buiding and scoping to get a better idea of whats going on. Unless you are designing, power grids , or big electical machines grids or Big RF (can you read a smith chart?) you wont be needing much more than some modest network analasys, norton thevenin, ohms law etc to get going basic builds. The water analogy breaks down outsire of RLC networks. One thing to remember: electons (or holes) don't wizz along conudctors or through semiconductors its the charge that moves. Water analaogy busted. Nearly all my piers an colleages agree thta the complex maths didnt help much in geting ration solutions desined an built and debugged. Learn Tax Law.I did a degree in electrilcal engineering with tons of maths almos none of which helped me desing low level analog measurment systems. I thought it funny that so little of the course was actual electronics. Its true that equipped with the foundations helps. Are you a maker or a wrinkley accaemic? The best edcuction comes from making duff circuits and work out how to fix them.Get a god dose of Horowitz and Hill.Its like French, you cant realy learn it from a book, you need to get out there and start trying it out. (I mean French, not the 16th gibberish they use in Quebec)..Get into the habit of thinking about current before voltage.Read areal world component dataasheet for an op-amp and decode what they are saying and why it will matter.If you want to go digital, just learn to count to one!You can make tons of interesting stuff without a micro or an arduino

author: T3sl4co1l
author_rank: Super Contributor
timestamp: February 17, 2021, 02:33:25 am
content: As an example to whet your appetite -- pi comes from the sine wave.As Benta noted, we could solve all of this in real time -- and I mean that literally, with real numbers. But differential equations are hard, so we'd prefer not to.Note, computers are good at hard yet mindless tasks; we can set them to cranking through solutions in this way, and with much more particular (complex, nonlinear) systems than we're talking about here. And so we have SPICE for example -- a simulation environment which steps incrementally forward in time, evaluating an arbitrarily complex system in the only tractable way left.But we generally prefer to analyze things, when we can. Analysis gets exponentially (probably hyperbolically?) hard with problem scope, so if we find a simple enough method fits the problem, we'd be wise to use it!If we're concerned with LTI (linear, time-invariant) RLC networks, then we can apply such a simplification. In short, if we assume sinusoidal stimulus, then all the differential equations can be rewritten in terms of sums of sinusoids: we don't need to carry around the sines at all, just represent their magnitude and phase at a given frequency. Which sounds like complex numbers, but isn't a huge motivation for them, yet.When we solve a (LTI) differential equation in the time domain, after some work, we find two facts always pop up:1. The differential equation can be rewritten in terms of an auxiliary equation, a simple polynomial of one variable.2. We get solutions in terms of exponentials:$$ e^{t / \tau_1} \left( A \sin \omega_1 t + B \cos \omega_1 t \right) + t e^{t / \tau_2} \left( C \sin \omega_2 t + D \cos \omega_2 t \right) + ... $$where, -- you'll have to excuse me as it's been YEARS since I did my diff eq homework -- the parameters (tau, omega) are in fact the roots of that auxiliary polynomial. Which, in general, involve algebraic numbers (sqrt and worse, or for real-valued inputs, whatever), including roots of -1, the imaginary constant.But in that case, or indeed, in general -- we don't need to write out the real and complex parameters, we can use Euler's formula,\$ e^{ix} = \cos \theta + i \sin \theta \$so we substitute an exponent of \$tau + i \omega\$ and discard the sin+cos stuff.And for further elucidation on that remarkable fact -- 3blue1brown on YouTube has one of the best visual explanations of how this works out.So, we get complex numbers, because the real part manifests as a damping (exponential decay) term, the imaginary part manifests as an oscillatory (sinusoidal) form, and the waveforms are systems of exponentials (e^t, t e^t, etc.). The aux. equation doesn't really seem to mean anything, but it sure is easier.Wouldn't it be great if we could just work with the auxiliary equation instead?Well, we sort of can. The framework built up around that usestransforms,of which Laplace has been mentioned, and is probably the simplest to start off with (however, the inverse is a bit ugly). The "full version" is the Fourier transform, which transforms a function of time to a function of frequency, the value of which is a complex number -- where B + iA has the A and B coefficients on the trig version.The definition of the Fourier transform, is to convolve the input function with every possible sinusoid -- "every possible" is done with an integral (or for the periodic form, the Fourier series, with an infinite sum of harmonics), and convolution effectively means to select for every sinusoid that correlates with the input.Also, I've been writing i for mathematical convenience, but I should really be using j, because us EEs already use i for signal current, go figure.Also also, this is all high level stuff, I think usually year 3/4 in EE? Or at least not 1st year. So it's being left intentionally vague, and will eventually fill in somewhat, if you can hold onto it until the next class that uses it and so on. Hopefully, your professors will give some proofs -- engineers tend not to, sadly, which to me at least makes the assertions, theorems, etc. much less impactful, and harder to understand. If this is the case, do take the time to challenge yourself through them, do extra homework -- it's quite worthwhile, at least if you're into that sort of thing.And also also also, as Terry said... math isn't for everyone, pretty much everything has been derived at this point, you only need to know what it is (for which, at least an introduction to the subject helps, if not a thorough understanding of it) and look it up in a book or whatever. Everyone has their range of interest, play to that.Like, me personally, I find network theory fascinating, but it's not an undergrad subject so I didn't get academic training in it (<-- BS EE), and honestly, working through some of those problems, like, trying to solve for some damn group delay or something (lots of arctangents, or 1/sqrt-something I forget which, because group delay has to do with... phase derivative, isn't it?), and it's all that work just to set up some equations that probably don't have an analytical solution anyway (in fact, even simple polynomials over order 5 have no closed form solution, let alone what kinds of equations are used to solve for other network properties) -- so it's a hell of a lot of work just to set up an equation that a computer has to solve anyway (or you with a LOT of grind work -- as they had to, back in the day -- network theory dates back to the 1920s, in particular some seminal papers in BSTJ, free on archive.org if you're curious) and then you scratch up a big old table of numbers,and you stillcan't quite use it yet because real components are inexact and a real circuit needs tweaking----- So, for the most part, we do use those hard-won solutions as a starting point, then we simulate from there, using realistic models, and other sources of data.Tim

author: rstofer
author_rank: Super Contributor
timestamp: February 17, 2021, 04:41:18 am
content: I'm not going to get into the complex bits but, sometimes, a picture is worth a bunch. I have attached a simple graph of sin(x) and sin(x+pi/4) where the second graph has a 45 degree phase shift.The graph isn't important, the site, desmos.com, is! You can do a lot of graphs at the site and a picture really is worth a bunch.I will also point out symbolab.com and it's ability to solve just about anything. And it will provide an explanation if you are a member. Well worth joining.

author: rstofer
author_rank: Super Contributor
timestamp: February 17, 2021, 05:03:52 am
content: And make friends with Python, MATLAB and or Octave (free MATLAB workalike). MATLAB (and Octave) are amazing solvers that easily deal with Laplace Transforms and just about every other math problem you will find in EE school.Python has some amazing libraries although I haven't used any for EE. I have been playing with machine learning and Python is absolutely the way to do the work. Great libraries!wxMaxima is another solver that is a big deal at CERNhttps://cds.cern.ch/record/1486286No, I don't have access to the document server but just knowing that maxima is used at CERN makes it worth the time to learn.And NASA still uses Fortran and I'm slowly getting up to speed with Fortran 90. It fixes a lot of the issues with Fortran IV that I started with in 1970. You may want to check into books like these:https://www.amazon.com/s?k=fortran+for+scientists+and+engineersNote the book on CUDA programming with Fortran. Fortran is the only language with built-in parallel programming features. You can break up your programs into parallel processes and run them on the CUDA units in your video card (NVIDIA and maybe others). CUDA units are very high speed floating point processorsDon't loop a thousand times, send the expressions to 1000 threads spread over many machines.Code:GPU cardCUDA cores RAMGeForce RTX 3080870410GBGeForce RTX 2080 Ti435211GBGeForce GTX 1080 Ti358411GBTesla V100 (16GB version)512016GBJust a few things to play with as you go along. The solvers will be invaluable.

author: edy
author_rank: Super Contributor
timestamp: February 18, 2021, 03:37:57 am
content: Thanks for all the feedback and explanations. I have the Art of Electronics book and several others as well, but this Stan Gibilisco book is the first one I'm reading that goes through the concepts in this particular order and has quizzes at the end of each chapter. I found it interesting that before even introducing the reader to even one practical electronics circuit or some examples of stuff to build, we are going through all this fundamental theory and the book is focusing on teaching all of this symmetry in the RLC maths. I think we covered current/voltage division through series and parallel circuits with respect to resistors, inductors and capacitors... for DC circuits, and then we started on AC reactances and the rest. I'm about 35% through the book so we'll see where it goes. I'll try not to get too hung up on the maths and just accept it for now and ponder it and let it stew... eventually I will internalise it and make sense of it. Once I get through it, back to reading Art of Electronics which I never quite finished last time I started it!

author: basinstreetdesign
author_rank: Frequent Contributor
timestamp: February 18, 2021, 05:34:36 am
content: Edy, if you want a low-stress way to understand all of those things about "relates to trigonometric functions and natural exponents, phasor calculus", etc. etc. Watch a lecture or two of Agarwals course at MIT on electronic engineering. His course is about 26 lectures, each about 50 min in length and he does an excellent job of taking the student from ground zero through transistor theory to op-amps with enough math to make it all make sense usually with a demonstration thrown in for grins. And he develops the math without introducing the concept of Laplace transforms which I thought was a minor miracle. I guarantee, the more you watch the more you will want to watch. Here is the first lecture.He may seem to rocket through the material but if you're paying attention you won't be left behind.