Flux linkage and flux cutting

----------------- No relativity needed and no change in the flux linkage law except to realise that the enclosed area may change with time.

Look at the integral form of the applicable Maxwell equation in both cases. It is often simpler in such cases. In Artley, "Fields and Configurations" Holt Rinehart etc, 1965 the integral form of the appropriate equation is given in Eq. 9-1-49 as: integral of E.dl around a closed path =- di [integral of B.ndS over the surface enclosed]/di t which reduces to the usual integral form for S fixed and the relation that you give follows. . This form then can account for the case of your solenoid (transformer voltage) or for your moving conductor (speed voltage) where the loop geometry changes with time. Assuming an N turn coil enclosing a flux (BA) then this form can be reduced to V= - A(dB/dt) +B(dA/dx)(dx/dt)

For most cases where Maxwell's Laws in the point form are used, there is no problem. In machines, with a mix of air gaps and iron, one can go from the integral form to more convenient flux linkage forms V= d(L11i1)/dt

+d(L12i2)/dt +.... and further separation into d(L11i1/dt) =L11(di i1/di t) +i1(di L11/di x)dx/dt etc. followed by axis transformations so things can be expressed in terms of equivalent circuits with speed dependent sources and constant inductances. Krause & Wasynczuk, "Electromechanical Motion Devices" McGraw Hill, 1989 use this approach

Don Kelly snipped-for-privacy@shawcross.ca remove the X to answer

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

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).

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

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.

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

| 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).

------------------

I believe that you have the correct form, as well as the concepts involved, for the voltage induced. Pages 400-401 of Artley "Fields and Configurations" Holt Rinehart & Winston, 1965 gets to the same place by starting from Faraday expressed as V=d(N*flux)/dt (and then goes on to the point form that you show. I do think that this "variable contour" is something that would be seen except in machines where part of the system moves with respect to another. While the point form of Maxwell's equations do not appear to specifically consider the contour change (how could it?), use of the point form would actually account for motion in such cases as signals sent to a moving target. It is just that the integral form is more tractable for machines.

If you review any energy conversion, flux linkages rather than flux cutting are used. Flux cutting is inadequate to represent any but very simplified models.

While I don't get the partial derivative sign in plain text - it is obvious what is meant.

Don Kelly snipped-for-privacy@shawcross.ca remove the X to answer

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

---------- If you want to change the effective area of the loop, why not rotate it on a pivot so that the area presented to the field is position dependent? This changes the contour as seen looking in the direction of B. The loop can be square so that at any position a rectangle will be presented to the flux rather than an ellipse to simplify calculations. Do it with B constant and do it with B varying with time. In the first case there will be a voltage whose frequency depends on the speed of rotation while in the second case there will be sum and difference frequencies. It's not the same as you propose in that B.n is varied rather than dA but B.ndA will be the same.

You could do it at quite low frequencies - Use B at 60Hz and a coil rotating between 0 and 3600 rpm will not run into wavelength effects. It might be better to fix the coil and rotate the field as then the electrical contacts will be fixed. --

Don Kelly snipped-for-privacy@shawcross.ca remove the X to answer