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