The Clarke and Park transformations (Episode 8)

hi and welcome back to understanding motors last episode we talked about how changing the way you perform pwm switching can affect the efficiency and dynamics of your commutation however the commutation schemes we've developed thus far are not the best methods of commutation we could use but before we can jump straight to the ideal methods we have to develop some tools so that we can better understand them so let's get into it we're going to start today by briefly going back and changing some of our notation in episode 5 when we were talking about the magnetic field alignment method of torque we learned that in order to maximize our torque produced in a brushless motor we want our induced magnetic field to be orthogonal to and leading our rotor's magnetic field in the notation i used that episode the magnetic field vector generated by the stator of the brushless motor will be 90 degrees counterclockwise from the current vector as shown in the motor winding diagram however after releasing episode 5 it was clarified to me that this notation of having the current and magnetic field shown as perpendicular is not the standard method of teaching within the electrical engineering community and because my ultimate goal here is to provide you the viewer with a reliable and easily intuitive explanation of these topics i read up on it and both because i don't want my videos to not resemble what you see in your textbook and because i genuinely think the more standard notation is better for understanding the transformations we're talking about today than the methods i was taught i've decided to shift my notation i apologize if this change causes anyone to be confused but i'm gonna do my best to keep things as clear as possible in previous episodes we had shown our brushless motor like this and our y circuit like this for the analysis of six block commutation as well as everything else we've talked about so far this is completely fine but now we're going to change the look of this y circuit a little instead of representing the phase as a resistor which generates a magnetic field perpendicular to the direction of current we're going to change this to a coil of wires which generates a magnetic field in the same direction the current runs the reason we're actually doing this is because it's helpful to have our magnetic field vectors and our current vectors in line with each other because at the end of the day it doesn't really matter which direction the current is physically running just the direction of the magnetic field induced by that current flow however in terms of measurement and control it's easier to think about current running through phases than magnetic field generated now with this new notation a current into phase a will correspond to both a current vector and a magnetic field vector strictly to the left a current into b will produce vectors 60 degrees south of east and c will be 60 degrees north of east so now that we've adopted this more standard depiction of the diagram let's take a second to think about it the first thing i want you to notice is that this is a two dimensional diagram i know that this is a completely obvious observation but it's also a very powerful fact because our three-phase current and magnetic field vectors can be described on this two-dimensional plane it's possible to describe the result as a 2d vector and then we could theoretically generate any equivalent resultant vector from just two phases and this is the idea behind the clark transformation first implemented by edith clark who by the way was america's first professionally employed female electrical engineer the clark transform describes the move from the a b and c windings to the alpha beta frame we can largely derive this transformation geometrically seeing that a points strictly in the alpha direction b points in the negative cosine 60 alpha sine 60 beta and c points in the negative cosine 60 alpha negative sine 60 beta direction the clark transform also includes an external two-thirds multiplier and this keeps the vectors equal magnitudes on either side of the transformation i find that this idea is not super clear at first but a quick example helps to show why it's necessary if i wanted to run one amp strictly from right to left through the three-phase diagram it will go into a then because this is a balanced system which obeys kirchhoff's current law it would need to come out of b and c in equal proportions if we sum this geometrically this one amp sort of gets counted twice it gets counted once on the way in through a and then because of the geometry it gets counted another half of the time when it's coming out through b and c however in our alpha beta representation we're just talking about the actual current running in each direction so we'll need to take two thirds of this current represented by the summation of the abc frame to get one amp and just so you aren't confused if you see it there's actually two forms of this transformation the one i'm using here which is the vector magnitude invariant version and another version used for power analysis which is the power invariant version and it uses the square root of two-thirds instead of two-thirds so now we get what the clark transformation says but it can actually be further simplified because once again the three-phase system we're talking about is assumed to be balanced and thus it follows kirchhoff's current law meaning the current in phase a plus that and b plus that in c must equal zero by moving some variables around and doing some substitution we can then see that the current in alpha is equal to the current in a whereas the current in beta is the current in b minus the current in c divided by the square root of three so now we can describe the direction of current and induced magnetic field using the alpha and beta axes but it may not be immediately obvious why this is helpful as we previously stated inducing a magnetic field perpendicular to the rotor's magnetic field produces torque meanwhile if we induce along the direction of the rotor's magnetic field it will sum with the magnetic field of the rotor thus either amplifying or weakening it well we just showed how you can describe the equivalent induction of a three-phase motor in two directions so now we're going to take this two axis representation and analyze it from the perspective of the rotor we will do this through what's called the park transformation we're going to start by creating another reference frame which will turn with the rotor by convention the axes of this frame are referred to as the direct or d axis and quadrature or q axis the direct axis points in the direction of the rotor's magnetic field whereas the quadrature axis is 90 degrees counterclockwise of it so a magnetic field induced in the positive q direction will produce a counter-clockwise torque meanwhile one induced in the negative q direction will produce a clockwise torque whereas a magnetic field induced in the positive d direction corresponds to strengthening the magnetic field of the rotor an induction in the negative d direction will weaken the rotor's field since the dq axis keeps the same origin as the alpha beta axis we can describe a transformation between the two as a simple rotation matrix for those unfamiliar this is basically just a matrix of trigonometric relationships which can take a vector or orientation described in the alpha beta frame and then describe it in the dq frame thus the current in the q direction is negative i alpha sine theta plus i beta cosine theta and the current in the d direction is i alpha cosine theta plus i beta sine theta where this theta value is the angle between the alpha axis and the d-axis okay so now we have these transformations in reference frames so let's look at what the actual implications are first of all if we want to optimize the amount of torque we're getting per current in which we usually do we can say that at any time for a non-salient pole motor we want our current to point strictly in the q axis direction note that if we're using a salient pole motor the optimal direction depends on some other variables and will typically lead our q axis a little bit i'll link a set of mit class notes that talk about this in the description below for people who are curious but to keep things simpler let's presume we're using a non-salient pull motor and let's run through our six block commutation scheme again using this diagram starting off in the center of hall sector 0 and connecting our phases appropriately we are initially perfectly aligned with the q axis and are optimally generating torque however as we move across the remainder of this hall sector our direction of current is no longer aligned with the q axis continuing on we see that throughout commutation we are only perfectly aligned with this q axis at the very center of each hall sector and as we get closer to the edges of the hall sectors more and more of our current points in the plus or minus d directions this causes the amount of torque we're producing to oscillate up and down and it creates the torque ripple we talked about in an earlier episode so now we have the clark and park transforms in our tool belt next episode we're going to take the ideas we talked about here and work towards developing a commutation method that will smooth this torque ripple out

Channel: Jantzen Lee

Views: 33,340

Keywords: park transformation, power electronics, electrical engineering, motors, understanding motors, bldc motor, bldc motor controller, alpha beta transform, dq0 transformation, 2/3, clarke transformation, jantzen lee, mechanical engineering, brushless motor, park transformation explained, field oriented control, field oriented control explained, field oriented control tutorial, FOC, motor control, motor driver, how do motor drivers work, डीसी यंत्र, DIY motors, makers, how does the

Id: mbJOxqxLkLE

Length: 9min 2sec (542 seconds)

Published: Thu Aug 20 2020