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| Poles and Zeros |
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| IconicPCB:
Without a through understanding of maths, graphical solutions and rules of thumb would not come into existence. Bode plots , stability circles. Smith Charts. various nomographs are a product of thorough understanding of maths describing a particular condition. Don't be scared by the problem...understand and appreciate it. |
| rstofer:
--- Quote from: b_force on June 15, 2018, 08:10:54 pm ---Yes, some algebra is needed to understand a few things, but in practice you basically always use the same kind of formulas. --- End quote --- That was the motivation behind Laplace Transforms, to solve Differential Equations using just Algebra. Great, so now you have the factored numerator and denominator but what does it mean? We know that when any factors of the numerator go to zero, the expression goes to zero and the expression represents the output. In other words, when any factor of the numerator goes to zero, the output is zero. We also know that when any factors of the denominator go to zero, the expression blows up (heads toward infinity because we can't divide by zero). The output tries to grow without bound. This is a pole - the point where a factor of the denominator goes to zero. That part is almost trivial given the ability to convert the schematic components into polynomials in S. And, no, nobody solves these equations by hand. That's why we have Matlab: https://www.mathworks.com/help/control/ug/analyzing-the-response-of-an-rlc-circuit.html But that doesn't address the art of getting the poles and zeros in the right place to achieve some result. All the above tells you is what you have, not how to get what you want. Or even what you should want. I'm not convinced that Laplace Transforms are understandable with just Algebra as a backround. Yes, they can be manipulated with Algebra, that's the whole point. But what they represent are some truly ugly Differential Equations and I'm not convinced those are understandable without something beyond Algebra. It was easy in the old days to plug the differential equations into an analog computer and just twiddle the coefficient potentiometers to get the desired result. Today we have to do that with Matlab Simulink and the potentiometers are 'virtual'. Works the same though... I wish the 'knobs' looked better on the attachment... And the Mass-Spring-Damper problem is the same as the R-L-C problem. Same equations, different constants. |
| T3sl4co1l:
Yes, you can't simply point to a component and say "pole" or "zero" -- apologies to Polish readers, of course :-DD -- it is a concept that is several levels of abstraction down. Example: Physical: you can point at a resistor (component). Abstraction 1: a resistor has resistance (electrical characteristic). Abstraction 2: resistances combine with inductances and capacitances to make complex impedances. Complex impedances vary as a function of frequency. Abstraction 3: when a rational approximation (i.e., a ratio of polynomials) is used for this approximation, we can factorize the numerator and denominator. When this is done, the factors are called zeroes (numerator) and poles (denominator). Abstraction 4: optionally, we might analyze how the poles and zeroes vary as a function of circuit parameters (e.g., component value, amplifier gain, etc.). In this case, we apply theorems like the Routh-Hurwitz stability criterion. As long as you are capable of grasping abstractions -- this shouldn't be challenge for you. A journey, certainly; there is a lot to learn here! In general, any analysis, done at this low level of abstraction, works in the mathematical domain of polynomial factorization. It's a notoriously difficult class of problems, so the mathematical tools are difficult to use, and the numerical results are typically unstable (i.e., small changes in inputs sometimes cause large changes in outputs). As engineers, that's okay for us; we're fine with "close enough", or "tweak until it's right". :) In that case, the hardcore mathematics isn't needed, and an easier lesson helps inform us what adjustments we should be targetting, so that the design isn't solved in a single step (which might be possible, but quite difficult), but rather, evolved until a close enough, practical solution is had. Tim |
| rbola35618:
There is some truth that knowing the mathematic is important but I think understanding the concept is more important. I am a power supply designer and I calculate the pole and zeros of the power supply's output filter circuit. To compensate the loop I use the concept of pole/zero cancelations. In other words, I calculate the pole in the output circuit and I then cancel it by placing a zero in my error amp at the same frequency. I then calculate the zero in the output circuit and then place a pole in my error amp at the same frequency to cancel each other. Here is a video where I use this pole zero cancellation to get a stable power supply. I show a graphical way of setting the loop and use very little math to accomplish this Skip to 28 minutes into the video where I show the graphical way of showing how the loop is compensated to pole/zero cancellation. I hope this helps make sense. |
| npelov:
... That's a lot of info to process... Thanks everyone. I'll read it and hopefully get back with better questions. |
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