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Re: [femm] Motion induced eddy currents

marctt2@xxxxxxx wrote:

Hi Dave,
Could you please give a written explanation of how to simulate traveling waves in the context of approximating the effects of motion and induced eddy currents in FEMM ? Also, can you give an intuitive explanation of the limitations of this approximation ?

Thanks.
Marc Thompson

------------------------------
Marc T. Thompson, Ph.D.

Thompson Consulting, Inc.
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Adjunct Professor of Electrical Engineering
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The program can mimic motion-induced eddy currents in one special case. This is the case in which the motion can be represented as a spatially sinusoidal distribution of currents that moves with a constant linear or angular velocity. An example of this sort of problem might be a long-stator linear eddy current brake, where a plate is braked as it moves past an array of permanent magnets. If one considers just the fundamental of the permanent magnet array (by taking the first non-zero term in a Fourier-series of the equivalent current sheet representation of the magnets), it becomes this sort of special case.

Usually, the phasor transformation used in the time-harmonic problems is (if we write it in terms of potential, but the same idea applies for any quantity that you'd transform):

A = Re[a*Exp[j*omega*t]] = Re[a]*cos[omega*t] - j*Im[a]*sin[omega*t]

Where "a" is a complex-valued number representing the phase and amplitude of A, "j"=Sqrt[-1] and "omega" is the frequency.

Now, say we had some sinusoidal distribution of current that is moving with a constant velocity:

J = Jr*cos[lambda*(x + v*t)] - Ji*Im[a]*sin[lambda(x + v*t)]

where lambda=Pi/(pole pitch) and the current is moving to the left with a velocity of v.

By comparison with the above phasor transformation, we can note that this could be represented as the phasor transformation:

J = Re[(Jr + j*Ji)*Exp[j*lambda*(x+v*t)]]

We can expand the exponential part to get:

J = Re[(Jr + j*Ji)*Exp[j*lambda*x]*Exp[j*(lambda*v)*t]]

What this means then is that we can represent this as a FEMM problem if we set the frequency to be: omega = lambda*v. We'd then approximately represent the region in which the current lives by breaking the region into a set of blocks, with the center of the nth block being centered at xn. The current density for the nth block would then be:

J[xn] = (Jr + j*Ji)*Exp[j*lambda*xn] = Jr Cos[lambda xn]- Ji Sin[lambda xn]+ j* (Ji Cos[lambda xn]+Jr Sin[lambda xn])

The amplitude of the current density is the same everywhere, but the phase is varying with position.

An example in which I did the same sort of thing in a rotating problem is http://groups.yahoo.com/group/femm/message/1260 This was a small PM motor where the rotor is a single permanent magnet. I'd approximated the eddy currents induced when the rotor was spinning by first thinking of the magnet as a sinusoidal distribution of current sheets and then making them "move" by picking the frequency to be equal to the rotational speed and varying the phase of the currents around the edge of the magnet like: J=Jr*(Cos[theta xn]+ j*Sin[theta])

Dave.
--
David Meeker
email: dmeeker@xxxxxxxx
www: http://femm.berlios.de/dmeeker