Coordinate transform

[Ryc92]

I am currently studying the use of a coordinate transform for the analysis of polyphase electrical machinery. I am having trouble understanding the transformation angle used in such.

For example, consider the transform of a three phase winding carrying current to any abitrary reference frame. In the stationary frame, the Q axis equivalent (using the transform shown in a book of mine, I know there are several different forms of it) component would be equal to:

Iq = 2/3 * (Va Cos(theta) + Vb cos(theta - 120) + Vc Cos (theta +120))

Now, if I choose to adopt a stationary reference frame in which the Q axis of such would coincide with the phase A of the real winding, theta would be zero right?

If I then choose to refer the winding components to a rotating reference frame, this angle would be changing at the angular speed omega, right? What would I then input as the angle? For example, if I choose to refer it to a speed of the excitation frequency (so in synchronism with the real winding angular frequency), what would I input as the angle theta to physically compute the values?

Unfortunately I see a lot of information on this topic on the internet that merely states the transform in matrix form, and does not show an example using real values (or perhaps I learn a little slower than others, lol).

Many thanks for any assistance you can offer, Ryan

 

[LiteYear]

I'm no expert at this, but let's see if I can help anyway:

Iq = 2/3 * (Va Cos(theta) + Vb cos(theta - 120) + Vc Cos (theta +120))

You've mixed current and voltage. While the transformation works for both, I don't think it will work when mixed. So the equation should be:

Iq = 2/3 * (Ia Cos(theta) + Ib cos(theta - 120) + Ic Cos (theta +120))

Secondly, this looks like the equation for the d component to me, not the q component. I'll assume your reference is non-standard but everything works out in the end.

Now, if I choose to adopt a stationary reference frame in which the Q axis of such would coincide with the phase A of the real winding, theta would be zero right?

Again you would normally coincide the D axis with phase A, not the Q axis, but again, this is just a convention. You can do what you wish as long as you're consistent! In any case, yes, you can choose theta equal to zero.

If you do choose theta equal to zero, the transform reduces to the alpha-beta transform, or the Clarke transform.

If I then choose to refer the winding components to a rotating reference frame, this angle would be changing at the angular speed omega, right?

You could do that too, but it would be a different transform. You can't mix and match.

What would I then input as the angle?

By convention, you would choose theta = wt for a rotating reference frame. Note that w (omega) can either be the synchronous speed or the rotor speed, depending on what you want to rotate with.

For example, if I choose to refer it to a speed of the excitation frequency (so in synchronism with the real winding angular frequency), what would I input as the angle theta to physically compute the values?

theta = w*t, where w is the excitation frequency. Note that if w is not constant things are a bit more tricky. Technically theta = integral(w, w.r.t t). If w is constant, then theta = w*t, otherwise you need to include the integral in your equations.

Does that help?

 

[Ryc92]

Hi, Many thanks for taking the time to reply.

The mix up of symbols is purely accidental, but thanks for noticing. I believe the difference in transform depends on whether the Q axis leads or lags the Direct, and I am glad to hear it is purely convention lol.

Sorry, I meant to write the next step in the transformation to a rotating frame:

IQ = Iqs*cos(theta) - Ids*sin(theta)
ID = Iqs*sin(theta) + Ids*cos(theta)

Where Iqs/ds refers to stationary frame variables

Does that look correct? Can I input wt (omega * time) for theta in this equation (I.E. can I use any speed I choose?)?

I am trying to build a circuit model for understanding the use of vector control in an induction motor in terms of orientation of field flux etc.

Again, Many thanks for taking the time to reply, Ryan

 

[LiteYear]

Yep, that looks correct. And yes, you can input theta = omega * time, but remember only if omega is constant.

Note that all that transform does is rotate two vectors, Iqs and Ids, anticlockwise by theta.