Machine terminology problem.

[jpcqub]

Having difficulty understanding a bit of machine terminology and I’m hoping you chaps might be able to help…

The main field of an alternator can be thought of as a single phase winding, producing one field travelling in one direction and another field travelling in the opposite direction.

Discuss?!?!

 

[electricpete]

I would say the statement is true if you consider only mmf from the stator current.

A field oscillating along a linear axis is roughly equivalent to sum of a forward and backward rotating field.

The proof is shown as follows:

Forward rotating wave is
Forward = cos(w*t + p*theta)
Applying cos(a+b) = cos(a)cos(b)-sin(a)sin(b) we have
Foward = cos(w*t)cos(p*theta) -sin(w*t)sin(p*theta)

Backward rotating wave is
Backward = cos(w*t - p*theta)
Applying cos(a-b) = cos(a)cos(b)+sin(a)sin(b) we have
Backward = cos(w*t)cos(p*theta) +sin(w*t)sin(p*theta)

Add the forward and the backward waves

Forward + Backward = 2 * cos(w*t)cos(p*theta)
This result have a shape in space which is fixed over time (cos(p*theta)). At any point it varies sinusoidally. It is a stationary variation, not a rotating variation.

What was the question?

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[electricpete]

The context in which this relationship is well known is the starting of single-phase motors. A single-phase motor winding by itself generates only pulsating field along the linear axis between pole pairs. The field does not rotate. Rotor may have difficulty starting but once it is given some help, the linear pulsation is enough to give pulsating torque to the rotating rotor. To overcome this we have various extra gadgets in single-phase motors - auxiliary winding, starting capacitor, shaded pole. These features are only required to start the motor and no benefit once running. The cap or aux winding are sometimes removed by centrifugal switch after the motor starts.

For three phase motors, each phase has forward and reverse rotating phase. But when you add the three phases together, the forward components add and the reverse components cancel so the stationary stator winding is capable of generating a true rotating field (without those extra starting gadgets).

Notice I focus on the field associated with the stator. If you add the rotor field into your question, clearly it is rotating.

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[electricpete]

I could have eliminted a lot of trig by starting with trig identity:
cos(a)cos(b) = 0.5*[cos(a+b)+cos(a-b)]

This leads directly to:
cos(w*t)cos(p*theta) = 0.5* [cos(w*t+theta) + cos(w*t-theta)

LHS does not rotate.
RHS is sum of forward and backward rotating field.

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[Skogsgurra]

Here's the second part of the explanation:

Pete has demonstrated that a single phase sinewave generates two rotating fields with opposing directions. I guess that the next question is "how can two opposing fields generate torque in a rotor?"

To make things simple, consider the rotor of an induction machine having a resistance and an inductance. The magnetic field induces a current in the rotor and the current is proportional to frequency divided by rotor winding impedance (Ir = k*f/sqrt(R^2 + X^2) or Ir = k*f/sqrt(R^2 + (2*pi*f*L)^2)

Torque is the result of magnetic field times rotor current (Ir) times cosine(alpha), where alpha is the phase-shift between magnet field and Ir.

Looking at the two rotating fields and the Ir induced in the rotor, you will see two identical currents that produce identical, but opposing, torques. So the net torque is zero.

If the rotor is set in motion in one direction, the Ir frequency corresponding to that direction decreases and so does the phase shift. The current magnitudes remain approximately constant as long as X >> R. The result is a net torque working in the same direction as the rotation, and the motor accelerates.

When it gets up in speed, the X part goes down even more and the phase shift between field and rotor current gets close to zero. At the same time, the opposing current stays nearly constant (double frequency makes R very small compared to X and the opposing current is given by k*f/(2*pi*f*L) or k/(2*pi*L) with a 90 degree phase shift.

This means that the "constructive" torque is proportional to slip and that "destructive" torque is essentially zero - or very close to.

This is the (quite verbose, admitted) explanation to how a single-phase induction motor (with or without an auxiliary winding) works. If it has an auxiliary winding, it can be switched off once the motor is running.

Gunnar Englund

http://www.gke.org

 

[electricpete]

That is great info Gunnar. I never thought about it that way.

I note a paradox in the equations. I think I know the answer but I'd rather call it a quiz for you guys:

Ir = k*f/sqrt(R^2 + (2*pi*f*L)^2) where f is slip frequency.

This suggests that rotor current is lowest at low slip and steadily increases as slip increases until it flattens out at very high slip at max value k / (2*Pi*L). So we would conclude that the reverse current at full speed (slip = 2) would be as high or higher than the current during starting (slip = 1).

Can you spot the flaw in this argument?

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[electricpete]

I was thinking the flaw was in development of equivalent circuit model, in particular the step where you divide rotor voltage and impedance by s. But now I don't think so anymore. I can't find the error in the logic above which suggest reverse rotor current is very high while running.... a conclusion that I know is incorrect.

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[jpcqub]

Thank you for your speedy and informative replies. I’m now reading up on revolving-field theory….interesting!

Thanks again

 

[Skogsgurra]

The reverse current is high, yes. But the phase angle is very close to 90 degrees (X very high in comparison to R) and that makes the braking torque low.

Gunnar Englund

http://www.gke.org

 

[electricpete]

I understand about the torque. But the current seems kind of strange.

I am trying to put together a picture of how large is that reverse current.

During initial start I have LRC. That includes a forward component and a reverse component. There is no symmetry which favors forward or reverse at the moment of start so I assume the forward and reverse components of rotor current each cause approx LRC/2 worth of stator current (neglecting magnetizing current which is typically much less than LRC... at least based on my experience with 3-phase motors... never worked with 1-phase much).

So now let's say I start the motor and get to full speed and run the motor unloaded. My reverse rotor current is still as high or higher than during start (as discussed above approaches k / (2*Pi*L)). So do we see stator no-load current of single-phase motors approaching LRC/2?

I don't know but I don't think so. I must be missing something. I realized single-phase motors were less efficient for a variety of reasons but I didn't realize they had that large of a reverse current.

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