EEVblog Electronics Community Forum
Electronics => Beginners => Topic started by: npelov on June 15, 2018, 04:07:25 pm
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Hi,
I can't grasp the idea of poles and zeros. Can someone explain to me how poles and zeros can be used to tell if a circuit will oscillate or not. For example here:
http://www.analog.com/en/analog-dialogue/articles/techniques-to-avoid-instability-capacitive-loading.html (http://www.analog.com/en/analog-dialogue/articles/techniques-to-avoid-instability-capacitive-loading.html)
poles and zeros are used to calculate the components needed for op amp compensation with capacitive load. What is pole frequency and zero frequency?
All the videos in youtube are about mathematics. I can't translate that to electronics. For example "stable" system is one that goes to "0", "unstable" goes to infinity and "marginally stable" goes to a constant value OR oscillates. When I say stable in electronics I imagine constant value and unstable when it's oscillating we look for the two cases of "marginally stable". Maybe I didn't understand correctly.
And why unity gain configuration is less stable than gain > 1?
Can someone explain and/or give me a link where these are well explained.
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This is basic EE, and if you want to understand it fully you'll need to learn the mathematics (which are normally called Complex Analysis).
Simply put, the poles and zeros are the solutions to the transfer function of the circuit.
The poles are the complex solutions of the denominator polynomium, the zeros are the complex solutions of the numerator polynomium.
In real life, you'll mostly run into second-order polynomiums, which can be solved relatively easily.
But you're working in the complex plane, and this is the first concept to get a grab on.
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All the videos in youtube are about mathematics. I can't translate that to electronics.
You won't be able to get away from that. Poles and zeroes are fundamentally mathematical in nature as they are components of the mathematical transfer function of a system. The practical mathematics is, contrary to initial appearances, quite simple to use as much of the application can be done in a formulaic fashion but, as with all mathematics, you need to understand what's going on before you start taking a 'fill in the formula' approach.
There are rules of thumb for determining stability that don't explicitly involve using poles and zeroes e.g. the Bode plot of the loop gain of a stable amplifier must cross the unity gain axis at a slope of -6dB/octave (-20db/decade). At the end of the day if you want to be able to cope with non-trivial systems then being able to handle the mathematics is almost essential or you get left with only empirical methods to use and that can be limiting.
I have struggled to find a good book that teaches control theory (for that is what this is) with a central focus on electronics that is more suitable for the, shall we say, mathematically challenged. I've found that too many texts either assume too much innate mathematical understanding on behalf of the reader, or are OK in that regard but are too focussed on general control systems and make scant mention of electronics applications.
If anybody has a book recommendation for a book that doesn't have these weaknesses I'd be very grateful to hear it as I always struggle to find a book to recommend to others (as in this instance).
And that's where I stop, as this really is the kind of area that needs you to sit down with a good textbook (or attend a good series of lectures). It isn't something that can be taught via a short series of posts on a forum - you might be able to get a flavour of the territory you're entering, and I suspect some others will make a few posts that might do that, but proper understanding is going to require a reasonable amount of effort studying.
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In real life, you'll mostly run into second-order polynomiums, which can be solved relatively easily.
He means polynomials.
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This is a beautiful book to start seriously learning circuit analysis:
Hayt and Kemmerly - Engineering Circuit Analysis
I suppose, that it is legal to download from archive.org the old 2d edition
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From the viewpoint of Control Systems (as opposed to circuits), this video series is excellent:
https://www.youtube.com/watch?v=gJzY6jOcgN0&list=PLmK1EnKxphikZ4mmCz2NccSnHZb7v1wV- (https://www.youtube.com/watch?v=gJzY6jOcgN0&list=PLmK1EnKxphikZ4mmCz2NccSnHZb7v1wV-)
This particular video just introduces the course and there are a few follow-on videos describing the class work before she gets down to it.
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Complex numbers, Euler's Identity, Laplace Transforms plus the predecessors (Calc I, II, III, Differential Equations) are probably required. Use of the Spirule is probably optional at this point because Matlab can do all these things.
That's why there are so many heuristic approaches. The math get completely out of hand. OTOH, it's easy to experiment with op amps and capacitors, less so for the control system of the unstable F35.
A quick example:
https://www.researchgate.net/post/What_is_the_physical_significance_of_Pole_and_Zero_in_a_transfer_function (https://www.researchgate.net/post/What_is_the_physical_significance_of_Pole_and_Zero_in_a_transfer_function)
See Figure 2 here:
http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf (http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)
It shows what happens to the output as the location of a pole is moved from quadrant to quadrant.
I think it was about here when I decided that I would work in electrical and not electronics.
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This is a beautiful book to start seriously learning circuit analysis:
Hayt and Kemmerly - Engineering Circuit Analysis
I suppose, that it is legal to download from archive.org the old 2d edition
The 8th Edition is there in two parts. Alibris has the 8th Edition as a paperback but it's pricey at around $80 new.
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In real life, you'll mostly run into second-order polynomiums, which can be solved relatively easily.
He means polynomials.
Thanks Cerebus. Of course. Sometimes switching between different languages leads you astray. :)
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This is basic EE, and if you want to understand it fully you'll need to learn the mathematics (which are normally called Complex Analysis).
Simply put, the poles and zeros are the solutions to the transfer function of the circuit.
The poles are the complex solutions of the denominator polynomium, the zeros are the complex solutions of the numerator polynomium.
In real life, you'll mostly run into second-order polynomiums, which can be solved relatively easily.
But you're working in the complex plane, and this is the first concept to get a grab on.
I never completely agreed with this answer.
Yes, some algebra is needed to understand a few things, but in practice you basically always use the same kind of formulas.
So instead of sending someone into the woods, I would rather explain what is import, what people need to look for and show them the basic tricks (which, let's be honest, are the only things you will ever need in real life).
Instead of scare them with extremely boring mathematics that 99% of the engineers are never EVER using anymore in their lifetime.
#howtowastetimeatyouruniversity
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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.
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Yes, some algebra is needed to understand a few things, but in practice you basically always use the same kind of formulas.
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 (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.
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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
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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.
https://www.youtube.com/watch?v=2doS4y9KsIM&t=1530s (https://www.youtube.com/watch?v=2doS4y9KsIM&t=1530s)
I hope this helps make sense.
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... 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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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.
One of my best teachers on my university for control theory had a much easier approach.
He just showed us a controlled valve system, what happens when it's underdamped, what happens if it's over damped and what happens when these "poles and zeros" are placed wrong.
Understanding a problem doesn't mean you need to know all the math behind it.
It means that you understand why it happens and what needs to be done to fix it.
Some people think in theoretical math equations, other people think in physics.
When a system is unstable you only need to use always the same equations to get it fixed.
Where are these equations are coming from is maybe nice for a Sunday afternoon reading, but you are never gonna think of it ever again
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Here is what I think about mathematics in electronics. I'm not that good at math, but it never stopped me before. I try to learn the absolute minimum to keep me going. I like to do the math when I can, but I know my limits and I when I reach them I learn more or I just find a shortcut. For practical engineering I this the truth is in the middle. There are people smarter than me who will do the theoretical part.
So I always try to simplify the problem. Do I need to know how to use poles and zeroes to make a feedback work - no. I know that an opamp can't immediately drive a capacitive load. It takes time to charge the capacitor. So when I see the solutions I have a basic idea why they put a resistor on the output and/or a capacitor in the feedback. If I overcompensate the opamp it'll still work. Maybe I don't need such a fast response. But I still want to try and learn as much as I can for cases when I need a faster response and tweaking doesn't do the job.
And in the real word you still need some experimenting and tweaking anyway. Theoretical part relies too much on you knowing every single detail about your parts. You may know the series resistance of a capacitor or the DC resistance of inductor but if you build a complex schematic with many active elements - opamps, transistors etc, the calculations are just the beginning, you may still have to measure, teak or adjust your calculations.
So, back to the poles and zeros.
@rstofer - I've stumbled upon some these videos (Control Course) before. I'll watch the whole series when I have the time. Thanks!
@rbola35618 Thanks for the video it is helpful. I'll have to watch it few more times.
In the link rstofer suggested (https://www.researchgate.net/post/What_is_the_physical_significance_of_Pole_and_Zero_in_a_transfer_function) Tolga Soyata says that the transfer function can be the impedance of a filter. When I first heard of transfer function I thought it's the relation of a output voltage to the input - Vout = F(Vin). The impedance is just a part of this relation and in this case it forms a divider with the input/output impedances of connected circuits. That confused me a bit. So the impedance goes to zero or infinity, but the output voltage. How do we choose what is the transfer function?
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In the link rstofer suggested (https://www.researchgate.net/post/What_is_the_physical_significance_of_Pole_and_Zero_in_a_transfer_function) Tolga Soyata says that the transfer function can be the impedance of a filter. When I first heard of transfer function I thought it's the relation of a output voltage to the input - Vout = F(Vin). The impedance is just a part of this relation and in this case it forms a divider with the input/output impedances of connected circuits. That confused me a bit. So the impedance goes to zero or infinity, but the output voltage. How do we choose what is the transfer function?
I don't agree with Soyata here. The transfer function describes which output you can expect from a certain input. This can be:
Po/Pi, Uo/Ui, Io/Ii, Uo/Ii, Io/Ui, Po/Ui etc., you can combine whatever you want.
Normally you compare apples with apples, meaning voltage/current/power input-to-output transfer functions.
Bringing impedance into play relates to two-port network analysis, which then can be transformed to a transfer function.
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As a non-EE (software engineer) who has strayed into power electronics and had to start from zero on this same subject, I must agree with Cerebus and say there is no running away from this mathematical subject if you want to design circuits that have feedback loops. You may be able to skate by, but you will be mostly limited to copying reference designs or making small tweaks to one. I've found it to be an illuminating way to understand circuits and well worth the effort to do things that aren't a straightforward application of a datasheet example.
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Since this is a question I have had for a long time and not seen any clear answers (including in this thread), I would like to echo the question from the first post, and try to pose the question more clearly:
If you have a transfer function and you can locate the poles and zeros in this function (take that as a given), how do you use the location of these poles and zeros to interpret the behavior of the circuit? So for example, maybe there is a pole at 100 kHz. What, practically, does this mean for the circuit and for stability analysis?
Lots of suggested answers explain how you can find the location of poles and zeros, but this is just the application of mathematics, it does nothing to answer the question.
Can anyone point to a resource, maybe like a "3blue1brown" video that really explains what the poles and zeros mean, what you should deduce from them, and what your goal should be when you try to shift their position or eliminate them?
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This lecture series by Roberge is one of great ones, imo.
Watching it over and over finally made sense of op amp compensation.
His book is long out of print but worth it if you can find a copy.
https://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/course-videos/
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The transfer function doesn't just describes the present output versus the present input, it includes history. Whenever you see a term like 1/s, you immediately say "integrator" and integration includes history.
Nobody wakes up one morning and says "Hey, I'll bet writing transfer functions in terms of s will get me a solution". We kind of work up to it by, among other things, struggling through Differential Equations and hoping that there is an easier way.
Khan Academy has an excellent program on all levels of math. They also have a program on Electrical Engineering. For Laplace Transforms:
https://www.khanacademy.org/math/differential-equations/laplace-transform (https://www.khanacademy.org/math/differential-equations/laplace-transform)
People often complain that they are not good with mathematics. Khan Academy is a way to overcome that when taken in slow doses and working through the quiz problems. As I said earlier, the EE curriculum is packed with math courses. All engineering programs are. Math is the filter on whether you go into engineering or something else.
That MIT paper I linked earlier has a lot of good information if it wasn't for all the equations. So, read the thing for the description and worry about the math some other time. Pay attention to the big picture and skip over the details. Figure 2 is especially important:
http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf (http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf)
Read the 6 numbered conditions under Figure 2 to get a qualitative sense of what the poles do when they are on the left half, on the axis or on the right half of the complex plane. It's all spelled out. You can see where systems become unstable if there are poles on the right half plane.
Read the last sentence of Page 6.
The next issue is which circuit elements, in which configurations cause poles and zero and how do they get to the left or right half of the complex plane.
Empirically, this is why amplifiers oscillate and oscillators don't. A little tidbit I learned in a Circuit Analysis course a very long time ago.
More than you ever want to know about loop stability:
http://www.cds.caltech.edu/~murray/books/AM05/pdf/am08-analysis_04Mar10.pdf (http://www.cds.caltech.edu/~murray/books/AM05/pdf/am08-analysis_04Mar10.pdf)
I'll look around and see if I can find a decent example of a circuit with poles in the right half plane.
The idea of cancelling a pole with a zero simply means that both the numerator and denominator have a common factor which can be cancelled. From Algebra I, thankfully!
Bottom of page 43 gets down to it:
https://www.springer.com/cda/content/document/cda_downloaddocument/9783319279190-c1.pdf (https://www.springer.com/cda/content/document/cda_downloaddocument/9783319279190-c1.pdf)
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My control systems lecturer used to present a problem , speak of "gut feel" about the performance and then bring on the analytic solution and link the two.
Anyone who has gone through a graduate degree will recall the intense mathematical content of at least first two years of the course and how the tools provided by the maths subjects married into the engineering topics.
On the question of "1/s =history"
If above statement makes no sense just remind Yourself of natural and forced solution to a differential equation.
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That MIT paper I linked earlier has a lot of good information if it wasn't for all the equations.
I really like that paper. It is a model of clarity and certainly addresses my questions. Thank you for linking it.
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Basically, for any given opamp there is a fixed time delay between a signal being input and arriving at the output. This is mainly due to R/C time constants in the signal path, created by parasitic capacitance in transistor junctions etc.
Mostly we want less than the max gain, so we apply negative feedback.
As signal frequency increases, the inherent delay becomes a larger proportion of a signal cycle. When it exceeds half a cycle, the negative feedback becomes positive feedback.. and all hell breaks loose.
There are two ways to alleviate that; to ensure that the gain (of the amp itself) falls to unity or less before the critical frequency is reached, or to apply phase advance (differentiation) to the feedback signal to cancel out or reduce the timing error caused by the delays. Sometimes termed lag compensation, or lead-lag compensation respectively.
Since a unity gain application has the most feedback, this is where the problem most likely arises. It's not so much a question of stage gain as of whether the overall feedback is reinforcing instead of reducing, to HF signals. The more feedback, the more likely that will be.
Most general purpose opamps have a simple lag compensation capacitor which reduces the open-loop gain as frequency increases. This is actually too much compensation for a high gain application, so it's basically a compromise arrangement.
As to working out what values to use, sorry but that does require all those funny squiggles. Or else just try and see. Which I suspect is what's used in most cases. Square wave testing will typically show ringing or overshoots if the stage is undercompensated.
http://pbfcomics.com/wp-content/uploads/2016/09/PBF267-The-Breakthrough.png (http://pbfcomics.com/wp-content/uploads/2016/09/PBF267-The-Breakthrough.png)
HTH.
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The only article about negative feedback that I was able to comprehend: https://www.allaboutcircuits.com/technical-articles/negative-feedback-part-1-general-structure-and-essential-concepts/ (https://www.allaboutcircuits.com/technical-articles/negative-feedback-part-1-general-structure-and-essential-concepts/) .
Although, even after many hours reading articles on the subject and doing SPICE simulations and experimenting with a signal generator and oscilloscope I still nowhere near to be able to design, e.g., a power supply compensation network. Sadly, most EE textbooks I saw do not really cover opamp stability issues.
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Basically, for any given opamp there is a fixed time delay between a signal being input and arriving at the output. This is mainly due to R/C time constants in the signal path, created by parasitic capacitance in transistor junctions etc.
Mostly we want less than the max gain, so we apply negative feedback.
As signal frequency increases, the inherent delay becomes a larger proportion of a signal cycle. When it exceeds half a cycle, the negative feedback becomes positive feedback.. and all hell breaks loose.
<snip>
As to working out what values to use, sorry but that does require all those funny squiggles. Or else just try and see. Which I suspect is what's used in most cases. Square wave testing will typically show ringing or overshoots if the stage is undercompensated.
A good discussion of the problems related to poles moving as a function of frequency. Circuits that are stable at DC can go unstable as the frequency is increased.
I think those squiggles are the difference between engineering and technicianing (I know it's not a word!). Most feedback systems are not simple op amp circuits, we can't plunk down feedback capacitors. Lots of times there are hydraulics, motors, valves, temperature controls and other physical phenomenon that resist tinkering. The frequencies are lower but the lag is considerable. Just getting a PID loop correct can be a lot of knob twisting (not withstanding 'Autotune').
In school, we had two courses related to controls. One instructor worked on the controls for the rocket nozzles on the Atlas missile and the other worked designing controls for nuclear power plants. It isn't as easy as you might think to get the Atlas away from the launch pad without falling over.
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It isn't as easy as you might think to get the Atlas away from the launch pad without falling over.
I don't necessarily think it's easy. Imagine balancing a pencil vertically on the end of your finger, and then pushing the pencil upwards fast without it falling over. This is kind of what a rocket is.
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It isn't as easy as you might think to get the Atlas away from the launch pad without falling over.
I don't necessarily think it's easy. Imagine balancing a pencil vertically on the end of your finger, and then pushing the pencil upwards fast without it falling over. This is kind of what a rocket is.
Put a blob of clay on the sharp end of the pencil (top) and while the system seems more unstable (mass at a distance), the inertia at the nose makes it easy to balance on the palm of your hand.
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This lecture series by Roberge is one of great ones, imo.
https://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/course-videos/
Unfortunately, every video in that series comes up with "YouTube video could not be played". |O First time I've ever seen that message.
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You have to follow along where they discuss downloading the videos as MP4s or viewing with iTunes.
I just downloaded the first video and it seems to work fine.
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This lecture series by Roberge is one of great ones, imo.
https://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/course-videos/
Unfortunately, every video in that series comes up with "YouTube video could not be played". |O First time I've ever seen that message.
The playlist on YouTube is linked below. Unfortunately the content seems to be blocked in the USA. In which countries do people find it to be available?
https://www.youtube.com/playlist?list=PLUl4u3cNGP62in17jH_DiJMkCGNM6Xni- (https://www.youtube.com/playlist?list=PLUl4u3cNGP62in17jH_DiJMkCGNM6Xni-)
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This lecture series by Roberge is one of great ones, imo.
https://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/course-videos/
Unfortunately, every video in that series comes up with "YouTube video could not be played". |O First time I've ever seen that message.
The playlist on YouTube is linked below. Unfortunately the content seems to be blocked in the USA. In which countries do people find it to be available?
https://www.youtube.com/playlist?list=PLUl4u3cNGP62in17jH_DiJMkCGNM6Xni- (https://www.youtube.com/playlist?list=PLUl4u3cNGP62in17jH_DiJMkCGNM6Xni-)
I am now in Germany, here you get "This video contains content from MIT. It is not available in your country."
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All those videos can be downloaded from archive.org, there's a link to them from the MIT open course site linked above - http://archive.org/download/MITRES.6-010S13/ (http://archive.org/download/MITRES.6-010S13/). Also it's worth noting that the whole textbook for that course, written by the lecturer, can be legitimately downloaded from the course site https://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/textbook/ (https://ocw.mit.edu/resources/res-6-010-electronic-feedback-systems-spring-2013/textbook/). Looks quite decent but it might still need more mathematical background than some here would be comfortable with.
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The only article about negative feedback that I was able to comprehend: https://www.allaboutcircuits.com/technical-articles/negative-feedback-part-1-general-structure-and-essential-concepts/ (https://www.allaboutcircuits.com/technical-articles/negative-feedback-part-1-general-structure-and-essential-concepts/) .
Although, even after many hours reading articles on the subject and doing SPICE simulations and experimenting with a signal generator and oscilloscope I still nowhere near to be able to design, e.g., a power supply compensation network. Sadly, most EE textbooks I saw do not really cover opamp stability issues.
You need wade through this: https://www.youtube.com/channel/UCq0imsn84ShAe9PBOFnoIrg (https://www.youtube.com/channel/UCq0imsn84ShAe9PBOFnoIrg)
I don't know how it could be made easier, althoug 3blue1brown also has good material.
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I am now in Germany, here you get "This video contains content from MIT. It is not available in your country."
I ran into this as well in USA. Here is where I found them available.
http://teachingexcellence.mit.edu/category/inspiring-teachers/james-k-roberg-6-302-electronic-feedback-systems (http://teachingexcellence.mit.edu/category/inspiring-teachers/james-k-roberg-6-302-electronic-feedback-systems)
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Hey ,
Start with this (don't worry it's easy to understand :))
https://www.dropbox.com/s/5lcyzu70hdqcoeb/Engineering%20mathematics.pdf?dl=0 (https://www.dropbox.com/s/5lcyzu70hdqcoeb/Engineering%20mathematics.pdf?dl=0)
If you study every day , it will take you about a month to go through the book.
When you are finished with that , take on this book:
http://s1.nonlinear.ir/epublish/book/Circuit_Analysis_For_Dummies_1118493125.pdf (http://s1.nonlinear.ir/epublish/book/Circuit_Analysis_For_Dummies_1118493125.pdf)