The discussion revolves around the concepts of phasors and space vectors in the context of electrical engineering, specifically focusing on the representation of sinusoidal functions and their transformations. Participants explore the mathematical relationships and proofs related to these concepts, including attempts to simplify or prove specific equations involving three-phase currents.
Participants generally agree on the definitions of phasors and space vectors, but there is disagreement regarding their relationship and the correct approach to simplifying the given equations. The discussion remains unresolved as participants explore different perspectives and mathematical techniques.
Some participants mention specific mathematical identities and transformations, but the discussion does not resolve the assumptions or limitations inherent in these approaches. There are also unresolved questions about the implications of time dependency in the context of phasors.
Hey I'm back, sorry about the delay.milesyoung said:Say you have some complex number ##z = re^{j\phi}##. The ##re^{j\phi}## bit is its polar form, which you can think of as a vector in the complex plane with magnitude ##r## and angle (phase) ##\phi##.
The complex number ##\frac{3}{2}e^{j\omega t}## then represents a vector of constant magnitude ##\frac{3}{2}##, which is rotating counterclockwise in the complex plane with angular frequency ##\omega##. See here for an illustration.
That's exactly the result you would expect in the framework of space vectors as they relate to electrical machines.
The vector shows you what happens if you apply a balanced set of three-phase currents to the stator of a symmetrical machine: it produces a stator current space vector, which rotates CCW in the plane normal to the rotor axis of the machine. There's no actual current with any spatial direction, that's nonsense, but the stator current space vector is aligned with the magnetic axis of the resulting stator field, which makes it a very useful abstraction (and one of the many great properties of space vectors).
Yes, did you have a look at the Wikipedia page?tim9000 said:I see how the phasee shifted cos-es are adding up to 3/2 with the frequency omega. But the way w're representing it with e^jwt
with the circle on the left hand side, does that imply that the y-axis is actually the imaginary axis?
I did, it's what I based my assertion. It just seems odd to me that we would model this circular path that takes place in an imaginary and real set of axies, to represent the actual vector of the peak mmf in real life. I can accept it though.milesyoung said:Yes, did you have a look at the Wikipedia page?
There's nothing in the framework of space vectors that you can't also express with vectors and matrix algebra, see, for example, the equivalent Alpha–beta transformation, but the algebra of complex numbers is often preferred by electrical engineers. You can just think of the real and imaginary components as coordinates in a plane (the complex plane).tim9000 said:It just seems odd to me that we would model this circular path that takes place in an imaginary and real set of axies, to represent the actual vector of the peak mmf in real life.