Three phase phasor proof or simplification

[tim9000]

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).

Hey I'm back, sorry about the delay.
Ok this is really dumb question but:
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?
Cheers

 

[milesyoung]

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?

Yes, did you have a look at the Wikipedia page?

 

[tim9000]

Yes, did you have a look at the Wikipedia page?

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]

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.

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).