Waveform of Classic Electromagnetic Induction

[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.

 

[b.shahvir]

In this type of arrangement is induced emf influenced by flux cutting also or only rate of change of flux linkage?

 

[Baluncore]

In this type of arrangement is induced emf influenced by flux cutting also or only rate of change of flux linkage?

It is rate of change of flux linkage.
Draw the magnetic field of your magnet.
Draw a circle about the mid-point of the magnet with a radius equel to the distance to the coil.
Plot the rate of change of line density crossing that circle, around the circle, to get the induced voltage.

 

[b.shahvir]

It is rate of change of flux linkage.
Draw the magnetic field of your magnet.
Draw a circle about the mid-point of the magnet with a radius equel to the distance to the coil.
Plot the rate of change of line density crossing that circle, around the circle, to get the induced voltage.

I understand but aren't the side conductors of the coil also getting cut by the non uniform flux density sweeping across it and as a result also contributing to induced emf quantities? Just a thought.

 

[Baluncore]

I understand but aren't the side conductors of the coil also getting cut by the non uniform flux density sweeping across it and as a result also contributing to induced emf quantities?

It is the rate of change of the number of lines passing through the area of the coil. It has little to do with bits of wire, it is all to do with an area.

 

[b.shahvir]

It is the rate of change of the number of lines passing through the area of the coil. It has little to do with bits of wire, it is all to do with an area.

So if I may presume, as a hypothetical case for that matter, the side conductors of the coil are of significant length. Will the flux cutting have any considerable influence on the induced emf in this case?

 

[Baluncore]

So if I may presume, as a hypothetical case for that matter, the side conductors of the coil are of significant length. Will the flux cutting have any considerable influence on the induced emf in this case?

The area of the coil is defined by the wire. If a line of flux enters that area, at the same time as another line of flux exits the area, then no voltage will be generated.
You must forget about the local wire and see only the area of the coil defined by the windings.

 

[b.shahvir]

If a line of flux enters that area, at the same time as another line of flux exits the area, then no voltage will be generated.

The above statement is valid for uniform flux densities but in this case flux density is non uniform and obeys Inverse Square law. Hence the the no. of flux lines entering and leaving the coil sides may not not be equal at that instant. A resultant emf would be induced in this case due to flux cutting and may contribute to the emf induced due to rate of change of flux linkage.

 

[Delta2]

If the coil is made of thick wire (i.e. it has radius comparable to the radius of the coil area , and also radius big enough that we can't consider the variation of the magnetic field inside it negligible) then the calculations for the total EMF can become a lot more complex as it will be a weighted average of EMFs that are always given by Faraday's law.

 

[Baluncore]

A resultant emf would be induced in this case due to flux cutting and ...

When it comes to a coil, how do you define "flux cutting" ?

It is not the flux lines crossing the wire that is important, it is the derivative of the total number of lines that pass through the closed area of the coil.

 

[b.shahvir]

When it comes to a coil, how do you define "flux cutting" ?

A non uniform magnetic field when sweeping across the coil will also tend to cut the side conductors of the coil (assuming side conductors are of significant length).
A resultant emf will be induced in the side conductors of the coil due to this phenomenon (speed emf, dynamo effect). This emf would in turn contribute to the emf already being induced due to rate of change of flux linkage. This is my understanding.