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Ref frame transf when one of the ref frame speeds changes w/ time

Ref frame transf when one of the ref frame speeds changes w/ time

Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
There are some oddities which occur when we have a time-varying frequency argument f(t) of a function like cos(2*pi*f((t) *t).   What happens is that the instantaneous frequency of the sinusoid is nothing like f(t).  Product rule tells us the instantaneous frequency is w_instantaneous = d/dt(2*pi*f((t) t) = 2*pi*(f(t)*t)' = 2*pi*(f'(t)+f(t))'

The simplest example is ramp change in frequency as shown in attached slides 1 and 2.  You can see in slide 1 that even though f(t) is changing smoothly, the time waveform instantaneous frequency changes abruptly as we enter and leave the ramp.  The analytical proof of this unexpected (to me) behavior is shown in slide 2.

I have simulations results in the synchronous reference frame (ref fram speed w is w=2*pi*LF) that I want to convert to the rotor reference frame.  I thought I could use the equations shown on attached slide 3 from Krause, and simply use w=wre(t) where wre = 2*pi*RotorSpeed(t)*Poles/2 = radian speed of an equivalent 2-pole motor based on my computed simulation results.   Slide 4 seems to imply that it is ok for the reference frame w to be a time-varying function.  But it occurs to me that this might introduce unexpected or unwanted behavior similar to slides 1 and 2..

The bottom line question: do you think it is acceptable to use a time-varying frequency in the reference frame transformation?  i.e. in slide 3 one of the theta's would be represented by theta = w(t) * t where w changes over time.
 

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
Correction to remove extra ' at the end:
w_instantaneous = d/dt(2*pi*f((t) t) = 2*pi*(f(t)*t)' = 2*pi*(f'(t)+f(t))'
should've been
w_instantaneous = d/dt(2*pi*f((t) t) = 2*pi*(f(t)*t)' = 2*pi*(f'(t)+f(t))

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
Correction to the correction:
w_instantaneous = d/dt(2*pi*f((t) t) = 2*pi*(f(t)*t)' = 2*pi*(f'(t)+f(t))'
should've been
w_instantaneous = d/dt(2*pi*f((t) t) = 2*pi*(f(t)*t)' = 2*pi*(f'(t)*t+f(t))

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

Pete, please proof read before posting!!!

Bill
--------------------
"Why not the best?"
Jimmy Carter

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
These are 2 minor changes posted within a short period, so nothing rolled to the top of the list that wouldn't otherwise have been on the top of the list. I have a hard time imagining that caused anyone any annoyance, much less enough to warrant screaming at a 3-exclamation point loudness.  You must really have been interested in my post... what are your thoughts?

 

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

I probably overreacted to my own confusion.
Cut that back to two exclamation marks. grin.

Bill
--------------------
"Why not the best?"
Jimmy Carter

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
Bill - I didn't mean to be saracastic but I was reacting to a tone that at the time seemed out of proportion to the correcctions in this thread. But I expect your comment was broader than just one thread, so I will accept your input and try to do better overall.

=======================

In case it wasn't clear, the only thing that was corrected in this thread was the very end of the first paragraph of my original post...

(f'(t)+f(t))'  becomes (f'(t)*t+f(t))  

Everything else stays the same. Nothing changes in the spreadsheet.  

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
The expression finstantaneous = f'(t)*t+f(t) does lend some insight into the cause of this anomaly.  The f(t) part we expect.  The f'(t)*t is unusual since it depends on t which at first seems preposterous.  But stepping back, it makes some sense.  If I am evaluating the value of cos(f(t)*t) at t=t1 and f(t) is increasing, we can imagine that the whole cos curve is anchored at zero and being stretched to the right, and the part of the stretching curve that affects my evaluation of cos(f(t)*t1) is everything between t=0 and t=t1.

So if I were trying to do a simulation involving a ramp change in frequency of applied voltage, what kind of expression would be used for that?

 

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
Arghhh. I figured it out.  For the parameter theta in the transformation matrix, the few examples that are in the book  are computed as theta = w*t (and these are all constant reference frame speed examples), so I guess I ASSUMED for some reason it should be the same when w=wr.  But for time varying w such as w=wr, I obvioiusly can't use that... I will need to use theta = integral{w(t}dt  instead.  I was settling down to go to bed when that popped into my head.  I could not restrain myself from saying "Duh" out loud and getting up to run to my computer to write it down.

Sorry... "never mind!"

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
And the other question of how to simulate a ramp would have a similar solution:
1 - theta(t) = integral{w(t)}dt where w(t) is the specified frequency profile such as a ramp
2 - v(t) = Vmagnitude*cos(theta(t))

=====================================
(2B)+(2B)'  ?

RE: Ref frame transf when one of the ref frame speeds changes w/ time

I think that main reason for this confusion (for me also it was very strange when I first read this) is in your first sentance in this thread. In equation cos(2*pi*f(t)*t) you said frequency argument f(t). It is frequency only if f(t) is constant.  If it is not constant it is not frequency.

Milovan Milosevic

RE: Ref frame transf when one of the ref frame speeds changes w/ time

To make myself more clear i will give paralel with linear movement.
If distance can be desbribed by equation s= k*t then k is speed only if k is constant in time. If not than it is not speed. So lineary increasing speed (or in your case frequency) doesnt mean that k should increase lineary, and because of that graphs dont have sense.
 

Milovan Milosevic

RE: Ref frame transf when one of the ref frame speeds changes w/ time

(OP)
Thanks Milovan. I agree.  

To follow the linear analogy, my graph of the original post would be equivalent to saying I drove 1 hour at 30 mph and 1 hour accelerating from 30-60 mph, so I compute my distance travelled as 60mph*2hrs =120 miles surprise.    The correct way to find total distance is obviously by integrating velocity over time instead of multiplying total time by final velocity.

Thanks.

=====================================
(2B)+(2B)'  ?

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