Waveform of Classic Electromagnetic Induction

[Delta2]

The position of maximum flux change occurs approximately when the magnet pole engages the coil edge (this happens twice for each pole face = four times per rotation)

Hmm , I semiagree to this, I can't understand why it happens twice per pole per rotation, do you mean once as it engages it from the bottom edge and one from the top edge?

 

[hutchphd]

Yes: on the way in (toward closest approach) and on the way out. In between these approach events the flux is small and changes sign but the derivative preserves sign.

 

[vanhees71]

I don't know, whether one can find this in the literature. Maybe it's worth trying to calculate the em. (wave) field for a rotating magnetic point dipole. The magnetization should be something like
$$\vec{M}(t,\vec{x})=\begin{pmatrix} \mu \cos(\omega t) \\ \mu \sin(\omega t) \\0 \end{pmatrix} \delta^{(3)}(\vec{x}).$$
The Maxwell equations read
$$\vec{\nabla} \times \vec{B} + \frac{1}{c} \partial_t \vec{E} = \vec{\nabla} \times \vec{M}, \quad \vec{\nabla} \cdot \vec{B}=\vec{\nabla} \cdot \vec{E}=0, \quad \vec{\nabla} \times \vec{E} + \frac{1}{c} \partial_t \vec{B}.$$
Perhaps one can even solve this analytically by calculating the retarded potentials.

Then you can put your current loop/coil and simply calculate the magnetic flux going through and the EMF by taking its time derivative.

 

[hutchphd]

Except this is not a point dipole. And that is the crux of the confusion. Its really two monopoles.

 

[b.shahvir]

Hmm , I semiagree to this, I can't understand why it happens twice per pole per rotation, do you mean once as it engages it from the bottom edge and one from the top edge?

The waveform from the swinging magnet experiment in attached pdf may help to understand this phenomenon of double hump sine wave.

 

[vanhees71]

Except this is not a point dipole. And that is the crux of the confusion. Its really two monopoles.

Why that? I'd say it's a very much idealized model for a rotating permanent magnet by just approximating the magnetization of the extended true magnet by a point dipole.

How would you write down a true point dipole? How is it different from two close (fictitious) magnetic monopoles of opposite magnetic charge (in analogy to the electric case)?

 

[hutchphd]

How would you write down a true point dipole?

I don't want to here.
It is not infinitesimal and this is very much near field. Perhaps I misunderstand your intent but the poles must far apart compared to the distance to and size of the coil for the OP effect to show up.

 

[b.shahvir]

Just imagine it to be a primitive hand driven permanent magnet generator.

 

[hutchphd]

Your result will be somewhere in between the very double-humped output of your video and the textbook example of a centered short dipole in a flat loop (which has a simple smooth sinusoid).. Do you need to know the exact solution ? The physics is pretty clear from those two limits for me.

 

[alan123hk]

The waveform from the swinging magnet experiment in attached pdf may help to understand this phenomenon of double hump sine wave.

I believe this phenomenon can be briefly described as follows.

When the north pole of the magnet is approaching to the coil, the induced voltage is +A, and when the north pole of the magnet is leaving the coil, the induced voltage is -A.

When the south pole of the magnet is approaching to the coil, the induced voltage is -A, and when the south pole of the magnet is leaving the coil, the induced voltage is +A.

The above process occurs in one cycle, so when the magnet is rotating, the timing of coil induced voltage becomes $+A~ →~ - A ~→~0~→~ -A~→+A$

Am I right?

 

[hutchphd]

Am I right?

Except it doesn't quite go to zero except at the crossings.

 

[alan123hk]

Except it doesn't quite go to zero except at the crossings.

I think you are right. In theory, only zero crossings can occur. Although the slope may be small, the coil induced voltage will not remain zero even for a short period of time.

 

[Paul Colby]

I worked at a company in the late 90s that manufactured linear stepper motors as part of a wafer prober. Each motor had 4 rare Earth magnets. At one point they had manufacturing issues with these motors. One theory was perhaps the magnets varied in strength enough to cause the observed out of spec wobble. I wound myself a Helmholtz coil and extracted a motor out of a floppy disk drive. Spinning the magnets generated a sinusoidal voltage proportional to the magnetic moment of the magnet. The measurements were quite precise, very reproducible. I was able to rule out variations of magnet strength as the culprit. Those were fun times.

 

[b.shahvir]

The above process occurs in one cycle, so when the magnet is rotating, the timing of coil induced voltage becomes $+A~ →~ - A ~→~0~→~ -A~→+A$

Am I right?

Yes, but there will be 0 emf state whenever the poles align with the axis of coil ( as rate of change of flux linkage is 0) and also 0 emf state when the magnet poles are 90 deg wrt axis of the coil (again rate of change of flux linkage is 0). So the result should be
0 +A 0 -A 0 and 0 -A 0 +A 0 per cycle
I wonder what the waveform will look like.

 

[Baluncore]

and also 0 emf state when the magnet poles are 90 deg wrt axis of the coil (again rate of change of flux linkage is 0).

No, the flux is zero at that instant, but it is changing through zero.

 

[b.shahvir]

No, the flux is zero at that instant, but it is changing through zero.

Yes but this may be more suitable in case of speed emfs (dynamo) rather than statically induced emfs as flux density obeys inverse square law and the magnetic flux density linking the coil would be zero when magnet is vertical (90 deg wrt coil plane). So rate of change of flux when magnet is passing through 0 would be 0, hence V=0 at that instant.

 

[vanhees71]

I don't want to here.
It is not infinitesimal and this is very much near field. Perhaps I misunderstand your intent but the poles must far apart compared to the distance to and size of the coil for the OP effect to show up.

I still don't understand your objection.

I just want to have a single rotating magnetic point dipole. It's of course the superposition of two perpendicular dipoles oscillating with a phase shift of $\pi/2$ relative to each other. Thus it's the analogue of the analogous case of a harmonically oscillating point dipole (the Hertz dipole). I'm sure, you can solve this problem as easily as you solve the Hertzian dipole you find in any textbook. Then you have a model field valid everywhere. In the near zone you have the quasistatic approximation, but I guess it's not much simpler to get than the complete retarded solution. If I find the time over the weekend, I'll post my result.

 

[Baluncore]

So rate of change of flux when magnet is passing through 0 would be 0, hence V=0 at that instant.

As the polarity changes, the derivative cannot be zero because the value is changing.
The sine curve rises through the origin, at that instant the value; sin( 0 ) = 0; but the rate of change is; cos( 0 ) = 1;

 

[Paul Colby]

Then you can put your current loop/coil and simply calculate the magnetic flux going through and the EMF by taking its time derivative.

It’s actually simpler than that. If one looks just at the time harmonic fields, there are a set of reciprocity integral relations which relate fields between two physical solutions. For harmonic electric fields and currents, for example, one has,

$\iiint E_1\cdot J_2 dv =\iiint E_2\cdot J_1 dv$

Where fields 1 are due to antenna 1 only and 2 due to antenna 2 only. All other boundary conditions are held fixed. For the magnetic case the B field in the coil due to an applied voltage is constant over the volume of the magnet (assuming proper design). This allows one to express the generated voltage as an integral of the constant magnetization in the magnet. Sadly, I’ve lost all the details to antiquity.

 

[rude man]

The emf will be a very irregular - not at all a sine wave - due to the irregularity of magnetic fields surrounding a permanent magnet.

The one thing you can say is that the average emf will be $4fN\phi$ where $\phi$ is the maximum net flux in the coil (when the perm. magnet points along the coil axis), N = no. of turns, f = frequency of perm. magnet rotation in Hz.
EDIT: I meant the average emf over 1/4 rotation of the perm. magnet.
The avg. emf for any integer number of rotations is of course zero.

 

[b.shahvir]

Kindly check the output voltage waveforms in the attached pdf article. This was the double peaked sinusoids I was mentioning about. In my opinion, the rotating magnet arrangement would generate such a double peaked sinusoid output voltage.

 

[Delta2]

I am not so sure that the device given in figure (b) of page 3 of this paper is equivalent to the arrangement of the OP. For sure it is not the exact same thing.

 

[b.shahvir]

I am not so sure that the device given in figure (b) of page 3 of this paper is equivalent to the arrangement of the OP. For sure it is not the exact same thing.

Both arrangements obey laws of EM so why should output waveform patterns be any different in rotating magnet case?

 

[Delta2]

I am not sure what you trying to say. The universe as a whole obeys the laws of EM (unless we going to dispute the invariance of the laws of physics in this thread which I don't think we should). Does this mean that every possible electromagnetic arrangement, every possible electromagnetic system in the universe that has a voltage output has the same double peaked sinusoidal voltage output?

 

[b.shahvir]

Does this mean that every possible electromagnetic arrangement, every possible electromagnetic system in the universe that has a voltage output has the same double peaked sinusoidal voltage output?

In my opinion it should, especially if the arrangements are quite similar barring the speed emf arrangement (flux cutting). There would be some valid explanation if otherwise which I am interested in.

 

[rude man]

Kindly check the output voltage waveforms in the attached pdf article. This was the double peaked sinusoids I was mentioning about. In my opinion, the rotating magnet arrangement would generate such a double peaked sinusoid output voltage.

I think we have a problem of semantics.
.
None of the signals I saw in that paper (except one) looked anything like a sinusoid.
A sinusoid is $sin(\omega t)$ or $cos(\omega t)$ or linear combinations of both (i.e. $A sin(\omega t) + B cos(\omega t)$.

It takes very carefully-machined pole pieces to design a (close to) pure sine voltage. At any given moment, $d ~phi/dt$ must be expressible in those terms: $emf = d\phi/dt = \omega~~ sin(\omega t)$ etc.. A very exacting task. A permanent magnet arrangement such as yours falls far short of such a flux.

On the other hand, the waveform emf will have some resemblance to a sine wave, expressible as a Fourier series. There will be a + and a - peak just as in a sine wave. I think that was mentioned in a previous post.

 

[Delta2]

I think we have a problem of semantics

Yes indeed, I think in this thread we redefined the meaning of sinusoid as anything that goes positive, has one or more (local) maximums, and then goes negative and has one or more (local) minimums.

 

[Delta2]

In my opinion it should, especially if the arrangements are quite similar barring the speed emf arrangement (dynamo system). There would be some valid explanation if otherwise which I am interested in.

If the arrangements are similar maybe, but here the only similarity I see at the very top level is that we have to do with systems of moving magnets and coils. But in one the (only) magnet does rotational motion and at the other we have multiple magnets that do horizontal oscillatory motion.

 

[b.shahvir]

Yes indeed, I think in this thread we redefined the meaning of sinusoid as anything that goes positive, has 1 ore more (local) maximums, and then goes negative and has one ore more (local) minimums.

Ok then in the rotating magnet case will the waveform represent double peak patterns or the normal sinusoid?

 

[Delta2]

Ok then in the rotating magnet case will the waveform represent double peak patterns or the normal sinusoid?

I am not sure but I think it will have double peaks.