# Flux linkage and flux cutting

After all these years, I still get the feeling that I really do not understand Maxwell's equations. This was brought home again when I
watched the Mechanical Universe episode dealing with Faraday's discovery of magnetic induction.
The example given was that of a toroidal solenoid with a few turns of wire looped through the toroid. There is no field outside the toroid so there is not flux cutting of the wire in the loop. The induced emf is given by the rate of change of total flux through the loop. The Maxwell equation describing this is
del x E = -�B/�t.
So my question is: How do you go from Maxwell's equations to describe the emf produced by a wire moving through a magnetic field?
If you bring in the theory of relativity to explain that motion through a magnetic field is Lorentz transformed to modify the magnetic field into a combination of electric as well as magnetic field, then explain what happens to the flux linkage law given above.
Bill
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wrote:

I think that I am beginning to see the light.
The induced emf is the line integral of the electric field, E�ds, around a closed contour. Using Stoke's theorem, this is the integration of curl of E over the area bounded by the contour. The Maxwell equation says that the integrand is -�B/�t. We can move the time derivative outside the integral to get
emf = d/dt Integral(B�dA) over a surface bounded by the contour.
The differentiation has to include the way the contour changes shape as well as the integration of B over the surface.
Change in the contour shape contributes the flux cutting component that can also be interpreted as the relativistic contribution to emf as the contour moves through a magnetic field.
During my education, I never had to deal much with taking derivatives of closed line integrals with changing contour shapes.
I still am concerned with possible effects that I do not understand.
Bill
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If you haven't been dealing with electromechanical energy conversion it is likely that you haven't dealt with a changing contour. Even there, the quasistatic "circuit" approximation is used e= d/dt(Li) where L is position dependent (eg. in a salient pole synchronous machine both self and mutual reactances depend on rotor position).
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Don Kelly snipped-for-privacy@shawcross.ca
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wrote:

I have not dealt much with such things. Most of the designers of electrical machinery probably did not use Maxwell's equations. The concepts of flux linkage and flux cutting were developed, in part at least, to come up with ways to avoid Maxwell's equations.
In dc machinery, for example, flux linkage and flux cutting give the same result. Unfortunately, there are cases such as the toroidal solenoid I described earlier where only flux linkage works. The problem arises as to which to use. In the dc machine you do not want to use both flux cutting AND flux linking.
Bill
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You don't have to use flux cutting and, as you say, you don't want to use both cutting and linking. You can start with consideration of the field and armature windings, their position and their connection -in particular the effect of the commutator. and end up with a pair of equations: vf =Rf*if +d(Lff*if)/dt +0 dia/dt va = d(laf*if)/dt +Ra*ia +d(Laa*ia)/dt Note that the armature flux axis is perpendicular to the field axis so the armature current does not induce a voltage in the field. The individual coils on the armature are moving with respect to the field so there is a speed voltage induced in them but the winding is fixed with respect to the field. The commutator does a wonderful job in handling this. Now if you shift the brushes a bit, there will be some armature field coupling.
I remember Dr Fett of U of Illinois asking me about why the mutual inductances are not balanced and I blew it. I made sure that I learned why.
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Don Kelly snipped-for-privacy@shawcross.ca
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Because of this forum, I thought about this to the extent that I now think that I understand induction from first principles--Maxwell's equations. To explain, I am restarting from scratch.
Unfortunately, USENET, AFAIK, does not allow simple ways of expressing complete line integrals. I would be able to paste pictures of the equations in various formates, but most newsgroups do not like that kind of attachment.
The key Maxwell equation for electromagnetic induction is
curl E = -�B/�t.
I hope that the partial derivative symbol shows up on non-Mac computers. One notation often used for the dot or inner product of two vectors A and B is (A,B), and that typography will travel well over the internet. The induced emf in a closed loop is the complete contour integral around the loop.
emf = Complete contour integral of (E,ds) around the loop.
By Stoke's theorem, this is the (double) integral of the dot product (Curl E, dA) over a surface bounded by the closed contour. From the Maxwell equation, this is the integral of (-�B/�t,dA). Taking the time derivative out from under the integration process, we get
emf = -d/dt {surface integral of (B,dA) bounded by the contour}
Here, the derivative acts on how the contour shape Affects the integral as well as how B contributes to the integral. This, I believe, is the fundamental mathematical description of induced emf.
This description has some interesting consequences for some relatively simple cases.
Consider a loop flipping in a magnetic field. This describes how a simple generator works. The value of the induced emf is correctly predicted by either flux cutting or change in flux linkage. This equality is what leads some people to think that flux cutting and flux linkage give the same results.
If one looks at the derivative of the integral, B is constant so that no emf arises from the B within the interior of the contour. It is the change of shape of the contour that leads to emf.
On the other hand, consider the loop to be fixed in space while the magnetic field is rotated. The contour does not change shape, but the integral changes as B changes.
Bill
Bill
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| Because of this forum, I thought about this to the extent that I now | think that I understand induction from first principles--Maxwell's | equations. To explain, I am restarting from scratch. | | Unfortunately, USENET, AFAIK, does not allow simple ways of expressing | complete line integrals. I would be able to paste pictures of the | equations in various formates, but most newsgroups do not like that kind | of attachment.
Make a web page and post the URL. There are numerous places that host picture hosting for free. Or figure out the language of Wikipedia/Mediawiki and use that (just preview making a new page in Wikipedia and take a page capture of it to am image file).
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|WARNING: Due to extreme spam, googlegroups.com is blocked. Due to ignorance |
| by the abuse department, bellsouth.net is blocked. If you post to |
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The more I think about it, the more certain I become that
curl E = -�B/�t
is the governing equation. This in turn leads to the flux integral through a varying contour described before.
I think that it would not be difficult to devise an experiment to check it out. I am thinking of an ac driven field between pole pieces that produces a nonuniform field, say something like a parabolic intensity as a function of radius. Then a loop is centered perpendicular to the field but does not extend outside the field.
The trick now,is to figure out how to make a conducting loop that can vibrate radially. I am thinking of a conductive stripe around some kind of breathing balloon. Any suggestions on how to implement that would be appreciated. This geometry allows for easy calculation of the integrals including a circular boundary changing with time. The output should contain sum and difference frequencies.
Operating frequencies will be low enough to avoid any effects arising out of short wavelengths. For example, the balloon could be driven at about 33Hz while the magnet could be driven at about 50Hz.
Bill