Symmetrical Components From a New Angle
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- Опубликовано: 16 сен 2024
- This video explores symmetrical component theory in a way that is not presented in electrical engineering school. Starting with the history and building up 1 phase at a time, the math of symmetrical components is explained in a visual way, showing the connection back to basic geometry. Then the general behavior of symmetrical components for any number of phases is briefly reviewed. To dive deeper into symmetrical component theory, I recommend visiting the Geogebra simulators linked below and interacting with the simulations there. If Symmetrical Component theory has always seemed like a mystery, then manipulating these simulators can help build a more natural intuition for the topic.
Submitted for the Summer of Math Exposition Round 2.
Content by Nathan Kassees. Credit to Oscar Flores for making a lot of the animations here. Check out his page at manim.dev/@red....
Geogebra Simulators
Wave-Phasor Relationship - www.geogebra.o...
Transform system phasors into symmetrical components - www.geogebra.o...
Transform symmetrical components into system phasors - www.geogebra.o...
V0 and the triangle centroid - www.geogebra.o...
V1 and the Outer Napoleon Triangle - www.geogebra.o...
V2 and the Inner Napoleon Triangle - www.geogebra.o...
6-Phase Symmetrical Components - www.geogebra.o...
12-Phase Symmetrical Components - www.geogebra.o...
1-root(n)-root(n) Napoleon Triangles - www.geogebra.o...
For more background on this geometric method of finding symmetrical components, I recommend the textbook chapter below and the references therein:
“Chapter XIII: Determination of Sequence Quantities from Phase Quantities.” Symmetrical Components, as Applied to the Analysis of Unbalanced Electrical Circuits, by C. F. Wagner and R. D. Evans, Mcgraw-Hill, 1933.
The elegance of your phasors stunned me.
Thanks for all the effort you put into those animations.
Thank you! To make the phasors, I worked with an animator who was proficient with Manim, the python library developed by Grant Sanderson for his YT channel 3blue1brown. My animator is mentioned in the description.
This is really wonderful. Although my core background is in mechanical engineering, I do understand electrical theory from time to time, and I fit into the same category of being able to work with the system but not really understanding it intuitively. Animations, Clarity of ideas and approach as well as the overall Teaching was spot on. Thank you, much appreciated. 👍
Thank you, THANK you for tying this back to geometry. I have never seen the Napoleon triangle proof, nor the centroid one! One thing that didn't seem intuitive to me was why we rotate those Napoleon triangles 180° after finding them. Other than that, the extension to n-dimensions was the exact thing my brain was thinking about next! Maybe with this construction, 6-phase power is something achievable to study. Crazy to think that there are transformers that provide that.
Would be interesting to see a vector treatment of Scott T and Zig Zag transformers, as proper phase rotation can mean equipment either works or blows up!
It's clear you put a lot of work into this video. Great job
Delightful!
Thank you for offering this lecture, I learned lovely truths.
Extremely intuitive and interesting point of view. Congratulations on this wonderful resource. Cheers from Chile!
Perfect explanation I have ever seen.👍
Try explaining the per unit system. I think you could do an amazing job. 😊
Well done! This was a great video.
Very nice visual interpretation! Good explanations. Very rich. Thanks for sharing all the resources. At 5:40 of the video, there’s supposed to be a j beside the sqrt(3)/2, correct?
Doh! You’re the first person to notice that mistake. Good catch.
Please do a second video going deeper on symmetrical components of higher-than-3-phase polyphase circuits.
Awesome video! Thanks a million... saying I got the picture will be too bold a claim, but thanks to you I have a rather clarified concept as of what we are chasing with these sequences :) !
I’m a synthetic geometer and this video taught me the practical use of Napoleon and Van Aubel(and their ugly complex proofs)
Oh, please do give some credit to Galileo Ferraris, 1847-1897, professor in Turin. The IEEE (Institute of Electrical and Electronics Engineers, they are the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity) in their Milestone program unveiled a plaque in 2021 at the Politecnico di Torino for the Milestone “Rotating Fields and Early Induction Motors, 1885-1888” with citation:
“Galileo Ferraris, professor at the Italian Industrial Museum (now Polytechnic) of Turin, conceived and demonstrated the principle of the rotating magnetic field. Ferraris’ field, produced by two stationary coils with perpendicular axes, was driven by alternating currents phase-shifted by 90 degrees. Ferraris also constructed prototypes of two-phase AC motors. Rotating fields, polyphase currents, and their application to induction motors had a fundamental role in the electrification of the world.”
Apparently, there was a series of court cases about the inventor of the polyphase motor and obviously the validity of Tesla/Westinghouse's patent, when Ferraris was long dead. Tesla won the final court case, allegedly thanks to a statement by three colleagues who testified that he invented it in the fall of 1887, before Ferraris' publication. Tesla and Westinghouse obviously then got all the fame and money.
Stanley (the one who you mention) commented on the court cases. One of his comments stated apparently "I myself have seen the original motors, models, and drawings made by Ferraris in 1885, have personally talked with the men who saw these models in operation and heard Ferraris explain them at that date." - from a letter to the Electrical Review on March 16, 1903 [Electrical Review, Volume 42. 1903. Pg. 415
Well I am not an expert on the court cases and the statements, but at least the IEEE does recognize Ferraris.
And yes, I'd rather drive a Ferraris than a Tesla 🙂
There were a lot of people to consider for that scene, including Ferraris, but I had limited space and time. I intentionally said Tesla “patented” and not “invented” the two phase motor. The main reason I mentioned Tesla is because his association with Westinghouse influenced the wide adoption of that system.
@@nathank7569 Thank you.
Thanks
Hmm, so it's just a discrete Fourier transform? And it's useful because the DFT is phase-invariant, so anything else which is phase-invariant like a motor only depends on the relevant components?
Like, it's cool, I just wasn't sure what we're trying to get exactly
You’re correct, the system can be treated at phase invariant, and relative components provide the information needed to understand system state.
AC power calculus is just the beginning. The next step is to calculate elecronics like buck/boost/pwm and so on for driving brushless motors and understanding capacitors and inductors.
Sorry, I'm having some trouble understanding.
From what I understand, Way to quantify unbalance.
a. Vector sum for V0
b. Rotate then vector sum and average for V1
why rotate Vb instead of Va, and why rotate 180 degrees. I'm guessing rotating Vb gives V0 and Va gives V1.
Then going to 3 phase, from my understanding o symmetrical component, the "a" operator is use to multiply to rotate then sum average. So I guess it is use to find (b)? To quantify unbalance?
And the notation V0 is usually reserved for zero sequence and this video it means something else?
At 4:49 talking about V0 measure the distance to the N, "geometric center of the system" (is this where va,vb,vc intersects?), to the centroid "geometric mean of the triangle? Or the other way round?
Can explain what's the significance of the translation symmetry?
I'm kinda stuck here. Thanks
The scene around 4:49 is strictly discussing the geometrical interpretation of the zero sequence. For a triangle, the geometric center is the centroid I’m referencing. The centroid is where the center of mass would be if the triangle was a solid object. The zero sequence vector is the vector pointing from the centroid to the neutral. Now if I want to shift the triangle across the plane without changing its shape, I.e. translational motion, then I just move my centroid, which is the same as changing he zero sequence.
The combination of positive and negative sequence vectors determines the shape of the triangle. The zero sequence vector does not change the shape of the triangle, but it shifts the triangle center away from the origin. Therefore the combination of all 3 sequence components allows me to draw any arbitrary triangle anywhere in the plane.
I hope that answers some of your questions. Let me know if it doesn’t!
Amazing
So V0 is a measure to the center of the system. Would that be the 'Neutral'? Or if V0 was anything besides 0 there would be a voltage on the neutral. Trying to think of what this correlates to on the Power System. Great Video!
I find it's easiest to think about the physical nature of symmetrical components by how each component affects a 3 phase motor, which is also the reason they were invented in the first place. If the positive sequence makes a motor turn at X speed clockwise, then the same amount of negative sequence will turn the motor at X speed counter-clockwise. Since the V0 signal looks the same on all 3 phases (i.e. you never multiply it by the complex sequence operator a), the motor can't distinguish a rotational direction when only V0 is present, and so V0 is the component that doesn't turn the motor at all. Thinking about it in terms of vectors, if you start with 3 different vectors, and add the same vector V0 to all 3, this is geometrically equivalent to shifting the origin by V0, which is the same as applying basic translation symmetry. So mathematically V0 can be interpreted as a translation in the complex plane, and physically can be interpreted as a 3-phase supply that doesn't rotate a 3-phase motor. Another physical manifestation is that V0 creates circulating currents in wye-delta and zig-zag windings. More on that in a future video.
@@nathank7569 Would it be fair to say that adding a V0 vector away from the Origin reflects adding a DC bias to a circuit, such that the AC phases don't all rotate about 0 V?
@@SeeNickView that’s correct, DC is zero sequence.
@@nathank7569 These are the connections that were never made in my university classes. I don't think we even covered symmetrical components. Thanks for shoring this all up, coming from an EE in renewables
@@SeeNickView feel free to add me on LinkedIn (Nathan Kassees) if you want to chat about other industry stuff sometime. I’ve worked in utility distribution for 14 years, currently working on microgrids.
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