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are there current harmonics on locked-rotor motor if we neglect satura

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electricpete

Electrical
May 4, 2001
16,774
I am pondering the meaning of stator winding pitch and distribution factors and what they tell us about motors.

Here is one thing I believe. If we have a motor with the stator energized by 3-phase sinusoidal 60hz, and rotor locked to prevent movement. If we neglected saturation (perhaps energize the motor at lower voltage in physical experiment), I believe there will be no current harmonics. Does anyone agree or disagree.

Here is my reasoning:

We now have a pure linear system. The phase voltages are the inputs (independent variables). The outputs (dependent variables) are fluxes and currents... these are all determined by linear operations (addition, scalar multiplication, differentian integration), and the system itself is not time varying (even though state variables are time varying of course). For a linear system excited by single-frequency sources in steady state, all state variables will be at that same single frequency.

Now if we allow the rotor to move we have a time-varying system introducing non-linearity, creates rotor mmf harmonics at frequency rotor-bar-pass frequency +/- 1*LF and current at same frequencies. Likewise saturation would allow currents at odd non-triplen harmonics.

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I agree fully, epete.

Your reasoning is correct and it can be easily proved by doing the experiment you just described. There are many transformers available for three-phase welding machines and disconnecting the rectifier and feeding the low voltage three-phase to a motor with locked rotor sets the scene. Using a Fluke 41 or 43 or some other FFT analyser to measure current will only give you as much harmonics as there is in the voltage. And not even that since the higher frequencies see more impedance than the fundamental and get more attenuated.

I saw someone in here saying that induction motors always have harmonics, but that just isn't true. There is some magnetising current that usually is distorted due to the non-linear magetising curve of iron, but that effect is quite small in an induction motor. They do not run saturated and the air gap makes the magnetic path quite linear. The slot frequency is not even a harmonic since the frequency is not an integer multiple of the mains frequency. And the amplitudes are neglible, too.

Just out of curiosity; why do you think that there should be harmonics?
 
For transformers, the excitation current always has harmonic content, even when operating in the "linear" region of the magnetizing curve. Even if you are in non-saturated region, the current required to produce the exciting flux is not proportional to the flux itself.

It seems that a motor under locked rotor conditions would still have some non-linearity due to exciting current and therefore would not be purely sinusiodal.

Maybe.
 
Sounds like we're all agreement based on physical reasoning.

What got me a little twisted around was the definition of the winding factors associated with a stator winding:
kdp = kd*kp

They have two definitions:
#1 - They give the mmf space harmonics that arise from stator slotting, even in presence of sinusoidal input voltage.
ie #1: stator slotting => flux space harmonics harmonics

#2 - They give the current time harmonics that arise from mmf (or more precisely flux) space harmonics.
ie #2: flux space harmonics => current time harmonics.

Put #1 and #2 together and you might expect:
slotting => flux space harmonics harmonics => current time harmonics.

While #1 and #2 are true in general, I don't think we can't put together #1 and #2 to make that last conclusion.

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I disagree totally with electricpete and skogsgurra statements .
Gentlemen, whenever an electric machine has distorted magnetic flux distribution it will have harmonics induced.

I don’t understand the “scientific reasoning” of skosgurra “ there is some magnetizing current that usually is distorted due to the no-linear magnetizing curve of iron, but that effect is quite small in an induction motor”

It is almost impossible to create a perfect sinusoidal flux distribution with a discrete spaced winding, if we add the iron saturation curve and the irregular air gap reluctance due to the stator and rotor slots: can you explain how a perfect sinusoidal flux distribution could be reached.

I hope you know how to develop a Fourier Series equivalent to an irregular periodic wave.

 
aolalde - Thanks for the response. I appreciate your disagreement. As I have said before I very much value your opinion, so I would like to explore it further.

It may be that I have not clearly identified my assumptions. One important assumption was saturation neglected. It is a very artificial assumption - just for my own purposes of separating different effects. Also I have assumed the rotor stationary (locked/blocked) for the same reason.

I agree that #1 (above) tells us we have space harmonics of the flux. Flux is by no means sinusoidal in space.

Normally we expect by #2 that space harmonics in flux imply time harmonics in the voltage (current). While #2 is true in general, I don't think it applies to flux space harmonics created by the stator slotting itself (or by the rotor slotting if rotor is stationary).

The reason these particular flux space harmonics don't cause current time harmonics I believe is because of these particular flux space harmonics arose from Line Frequency (LF) currents and permeance variations which are stationary.

So imagine the flux evolving over time I believe it is something like StatorMMF_Movie.AVI at the following link:


(note that the statorMMF movie is ~750k and rotorMMF movie is ~ 1.5meg)

So when plotted mmf vs theta it is a stairstep with the steps moving up and down sinusoidally in time but not left and right. Pick any stator loop. Voltage induced is rate of change of all the mmf within it's span. Each span contributes a sinusoidal time-verying contribution at Line Frequency (LF). Induced emf is a sum of all these sinusoids at LF and therefore emf is also sinusoidal at LF.

Another way to look at the same question is to search for sources of non-linearity. I believe there are none under the assumptions of no saturation and locked rotor.

But if we set the rotor in motion as per rotorMMF movie, the stairsteps are also moving horizontally on the page and that does introduce time harmonics.

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aolalde, I think we all agree that the flux wave will not be sinusoidal because the winding distribution isn't sinusoidal (because it is laid in discrete slots).

What epete is claiming is sinusoidal current, and I agree with the others i.e. that as far as it goes, the current will be sinusoidal. Although there is a non-sinusoidal flux within the stator that is linking all the coils, I think all the emfs (at the various harmonics) will cancel in each phase because all three phase windings are physically symmetrical, and epete has applied the assumption that the iron is operating linearly.

My reasoning is: it is a standard result that if you have a 3-phase system where the winding distribution of each phase is sinusoidal (some sort of airgap winding perhaps), producing a perfectly sinusoidal rotating flux, then the induced emfs due to mutual inductances between the phases will all cancel.
Taking it a step further, if each phase is now placed in slots and is therefore no longer sinusoidally distributed though the three phases are still symmetrical (i.e. identical but displaced 120°elec around the stator), then assuming it's a linear system the winding can conceptually be thought of as separate superimposed sets of sinusoidally distributed windings, one for each harmonic. Superposition applies so all the mutually-induced harmonic emfs cancel.

So I seem to have arrived at the same conclusion as Pete, that the current harmonics in normal operation aren't due to slotting.
 
Your proposed initial condition is very close to that of a transformer except for the air-gap and slots. If saturation is eliminated, the magnetizing current certainly will look sinusoidal. That eliminates the time induced harmonics or those nonlinearities due to the time variation of a sinusoidal applied voltage.


I still have problem to eliminate the slot effect in the air-gap reluctance (the mmf space harmonics that arise from stator slotting), due to the field rotating at synchronous speed, I think that variable air-gap reluctance, will distort the magnetic field, introducing harmonics in the magnetizing current. Unless you skew the laminations to reduce that problem.
 
aolalde,

Look at it this way: Since we agreed that the rotor is locked and that the iron is not saturated, there are no non-linearities and there are no temporal changes in the electric end magnetic circuits.

There is a rotating field, yes. But we have assumed a linear system, so we are allowed to study each phase indepenent of the others (superposition principle). Then, each phase winding produces a staionary (in space) field that varies sinusoidally with time. Everything in the electric and magnetic circuit is constant and there are no non-linearities - as I said before - so there are no harmonics.

The slots and the distributed winding come into play only if the rotor moves, and the effect will not be very noticeable even then.
 
Fwiw - I just remembered the correct name to describe the system I am thinking of (rotor locked, saturation neglected): “linear time invariant” (LTI). Within that class of system, steady-state single-frequency (time) inputs can only give outputs at the same time frequency.

Linearity seems the simplest way to prove it but may not be satisfying. A discussion of the physical picture how we can have non-sinusoidal space distribution but sinusoidal time induced voltage: If we look at the simplified pattern of the flux I showed in the stator mmf avi file - It is a stair-shaped flux. It is very non-sinusoial in space, but the horizontal portion of each stair step moves up and down sinusoidally @ 60hz in time. The vertical breaks on the stairsteps remain stationary.

Between slot 1 and 2 the flux is A1*cos(2*Pi*LF*t+Phi1). (sinusoidal in time at 2*LF)
Between slot 2 and 3 the flux is A2*cos(2*Pi*LF*t+Phi2).
Between slot z and z+1 the flux is Az*cos(2*Pi*LF*t+Phiz).
etc

What is the voltage induced in a coil that spans 1-10? Add up contributions from 1-2, 2-3, 3-4, 4-5 … 9-10:
Vcoil = N * D /dt * [ A1*cos(2*Pi*LF*t+Phi1) + A2*cos(2*Pi*LF*t+Phi2)…. +
A10*cos(2*Pi*LF*t+Phi10).]
(Where N is number of turns.)

The quanitites within the brackets are all at line frequency (although different amplitudes and phases) and can be added by vector addition to give total flux at LF of known amplitude and phase ie Atot*cos(2*Pi*LF*t+PhiTot). D/dt just creates a 90 degree phase shift. Therefore the coil voltage is also sinusoidal varying at LF.

What is the induced voltage created by the series sum of coils in a phase? Add up the voltage of all the coils using the same method we just used. The sum of coil-voltage sinusoids at LF (with differering amplitudes and phases) is itself a sinusoid at LF (induced phase voltage is a sinusoid at LF).

Now if the rotor moves, it remains a linear system but it would now be time-variant. (No longer LTI). New frequencies will show up in the current including rotor principle slot harmonics at rotor bar pass frequency +/- LF.

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Comments: Reference:
J.K. Binns and E. Schmidt "Some concepts involved in the analysis of the magnetic fields in cage induction motors," Proceedings IEE, Vol. 122, 169, 1975
List of harmonic components developed in a cage machine:
1. Stator
1a. Stator mmf
1b. Rotor mmf
1c. Slot-ripple modulated by 1a.
1d. Saturation, giving slot-ripple sidebands
2. Rotor
2a. Slip-frequency currents representing the main-flux fundamental
2b. Components associated with 1a
2c. Slot-ripple due to stator teeth
2d. Saturation as in 1d.
2e. Currents induced by harmonic fluxes
Chief loss producing harmonics are those in 1c., but serious aggravations may be produced by such asymmetries as unbalanced windings and gap nonuniformity. In general, a harmonic effect in the rotor is reflected in the stator and re-reflected therefrom into the rotor in a cummulative armature reaction effect.
 
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