Voltage generation in a coil

[Adriaan Kragten]

There are two ways to explain how a voltage is generated in a coil. Assume we have a constant magnetic field in which a rectangular coil is rotating. The coil has two in parallel long sides and two in parallel short sides. The axis of rotation is lying half way the short sides and is in parallel to the long sides. The axis of rotation is perpendicular to the direction of the magnetic field. Assume that the position of the coil for which the coil area is perpendicular to the magnetic field, is called the zero position.

First way of explanation
The maximum number of field lines are flowing through the coil for the zero position. If the coil has rotated 90°, the coil is in parallel to the magnetic field and no magnetic field lines are flowing through the coil. The generated voltage is proportional to the speed for which the magnetic flux is changing. The change is zero for the zero position and maximal if the coil has rotated 90°. This is because the direction of the magnetic field with respect to the coil, changes at this point. It can easily be proven that the change of the number of magnetic field lines flowing through the coil is sinusoidal and that it has a maximum for a rotational angle of 90° and 270°. So the generated AC voltage is maximal for a rotational angle of 90° and 270°

Second way of explanation
A voltage is generated in a straight wire which is moving perpendicular to a magnetic field. The voltage is proportional to the strength of the magnetic field, to the length of the wire and to the component of the speed perpendicular to the magnetic field. A short side of the rectangular coil can be divided into a part left from the turning point and a part right from the turning point. The voltages generated in these parts are just opposite each other and so the total voltage is zero. The voltage generated in a long part of the rectangular coil is proportional to the component of the speed perpendicular to the magnetic field. This component is zero for the zero position and at a rotational angle of 180°. This component is maximal at a rotational angle of 90° and 270°. The generated voltages in both long parts of the coil are strengthening each other. The component of the speed perpendicular to the magnetic field also changes sinusoidal so the generated voltage varies sinusoidal.

So both ways of explanation give the same result. If a PM-generator has coils with an iron core, like for most radial flux generators, the magnetic flux is concentrated in the iron and almost no field lines are flowing through the air in the grooves which are used for the coils. For this kind of generator, it is therefore easiest to to use the first way of explanation for the generation of the voltage.

For an axial flux generator with two steel sheets with magnets at the inside and a disk with coils without iron in between, the second way of explanation can better be used. As there is no iron which concentrates the magnetic field, there is about a constant magnetic field in between two opposite magnets. A coil has two legs and an inner and an outer coil head. The voltage is only generated in the legs as only the legs are moving through the magnetic field. The coil heads are required to connect the legs but the length of the coil heads should be made a small as possible to reduce the Ohmic resistance of a coil. The average pitch in between the left and the right leg should be the same as the pitch in between two adjacent magnets because then the generated voltages in both legs are in phase. If the generator has eight poles, the pitch angle in between two adjacent magnets is 45°. This means that the average pitch angle in between two legs of a coil should also be 45°. But as a coil has a certain thickness, the pitch angle of the inner turns will be somewhat smaller than 45° and of the outer turns will be somewhat larger than 45°.

For a 1-layer, 3-phase winding, the number of armature poles must be dividable by four. The number of stator coils is 3/4 of the number of armature poles. So for an 8-pole armature we get six coils. The coils are laid in the sequence U1, V1, W1, U2, V2 and W2. The optimum positions of the heart of the coil legs are then respectively 0° and 45°, 60° and 105°, 120° and 165°, 180° and 225°, 240° and 285°, 300° and 345°.

[electrondady1]

any diagrams forth coming ?

[Adriaan Kragten]

any diagrams forth coming ?

What kind of diagram do you mean? A picture of a rotating coil in a constant magnetic field or a wire diagram of a 1 layer, 3-phase winding for an 8-pole axial flux PM-generator?

I will first add some comment to my original post.
 
A disadvantage of a 1-layer winding is that only half of the possible leg positions is used. Within each coil, there are two positions which aren't used. The remaining twelve positions can be used for a second winding with six more coils so with coils U3, V3, W3, U4, V4 and W4. However, this results in crossing coil heads and crossing coil heads make the winding much thicker. Therefore a 2-layers winding is normally not used for axial flux PM-generators.

Another disadvantage of the described 1-layer winding is that the legs of adjacent coils make an angle of 15° with each other. This means that the thickness of a bundle of wires must be rather small to prevent that two adjacent coils touch each other at a small radius. The thickness of a bundle of wires can be increased if the two adjacent legs of two different coils are taken in parallel to each other. This means that the angle in between the legs of one coil is now increased from 45° up to 60°. The average pitch of a coil in mm may now differ from the pitch in mm of the magnets at the pitch circle. So now the voltage generated in the left leg of a coil may be not exactly in phase to the voltage generated in the right leg of a coil. This results in some reduction of the total voltage generated in both legs but this is acceptable as now there is place for much more copper. I will make a wire diagram of such a winding for an 8-pole axial flux PM-generator.

[SparWeb]

Thank you Adriaan!


Here's a graphic of field lines passing through a coil.



The lines can take any path around the outside of the coil, but as they pass through, only the amount of field passing perpendicular through the coil loop counts.
When taken in combination, the field and the area closed by the coil define a "flux".
If the field strength varies across the area closed by the coil, the average field is what counts to make the total "flux".
If some lines are passing "down" like the image and another nearby magnet of opposite polarity pushes line through going "up", they cancel.
If more lines pass through (a greater flux density) then more voltage will be generated at the terminals of the coil.
If the strength of the lines remains constant, the field becomes static and NO VOLTAGE is generated.
If anything changes the number of lines passing through, ONLY THEN can a voltage be generated.
The amount of voltage is proportional to the flux change through the coil.

I have always preferred the flux and coil way to show how generators work, without denying the usefulness of the other method other folks prefer.
The basic concepts are simple enough that they can be explained and demonstrated with simple examples.  Simple experiments with easy measurements give results that match the theory.  The method is also powerful enough to give accurate results in complex situations, when vectors, trigonometry, and integrals are required.

[kitestrings]

Thanks Adriaan.  It's nice to learn, or re-learn some of the theory behind why thinks we've come to expect work they way they do.  ~ks

[Adriaan Kragten]

Thank you Adriaan!


Here's a graphic of field lines passing through a coil.

(Attachment Link)

The lines can take any path around the outside of the coil, but as they pass through, only the amount of field passing perpendicular through the coil loop counts.
When taken in combination, the field and the area closed by the coil define a "flux".
If the field strength varies across the area closed by the coil, the average field is what counts to make the total "flux".
If some lines are passing "down" like the image and another nearby magnet of opposite polarity pushes line through going "up", they cancel.
If more lines pass through (a greater flux density) then more voltage will be generated at the terminals of the coil.
If the strength of the lines remains constant, the field becomes static and NO VOLTAGE is generated.
If anything changes the number of lines passing through, ONLY THEN can a voltage be generated.
The amount of voltage is proportional to the flux change through the coil.

I have always preferred the flux and coil way to show how generators work, without denying the usefulness of the other method other folks prefer.
The basic concepts are simple enough that they can be explained and demonstrated with simple examples.  Simple experiments with easy measurements give results that match the theory.  The method is also powerful enough to give accurate results in complex situations, when vectors, trigonometry, and integrals are required.

I have used a rectangular coil because for the second way of explanation, a voltage is only generated in the parts of a coil which are in parallel to the axis of rotation. The direction of the magnetic field B is horizontal. I have added a picture of the coil as attachment.

[Adriaan Kragten]

I have made a drawing for an 8-pole PM-generator with a 1-layer, 3-phase winding which I have added as an attachment. I have used magnets size 30 * 20 * 10 mm at a pitch circle of 90 mm. You need 16 magnets for an 8-pole generator. You look at the coils so you only see the magnets glued on the back steel sheet. The magnets can be bought at Enes Magnets in Polen. The gross price is about € 2.50 per magnet so the total magnet costs are about € 40 (excluding transport).

The upper picture gives the coil configuration if the angle in between the legs is chosen 45°. The thickness of a wire bundle can be about 6 mm for this winding. The lower picture gives the coil configuration if the angle in between the legs is chosen 60°. The position of the wires has been chosen such that the coil pitch at the pitch circle of the magnets is exactly correct. The pitch at a smaller radius is somewhat too small and at a larger radius is somewhat too large but these effects can be neglected and the voltage generated in the left leg of a coil will be almost in phase to the voltage generated in the right leg of a coil. It can be seen that for this option, a thickness of a coil bundle of about 10 mm can be used and still have enough room in between two adjacent legs.

It might even be possible to give the coils the shape of the rotor of a Wankel motor. So in this case both legs are curved with the same radius as used for the outer coil head. Manufacture of such coils might be easier than manufacture of coils with straight legs.

[Bruce S]

any diagrams forth coming ?

What kind of diagram do you mean?

Adriaan;
There are those of us who can read a description or read a theory or a word based math equation and not be able to visualize it.
I am one of those people, I can read a blueprint or math equation and almost know all the levels of what going on.  Give me a word description of even a simple focal point and I'd be hard pressed to "see" what going on without a full break out of each individual part of it.

This may have been what electrondady1 was asking too.

I basically deconstructed your post one part at a time to be able to wrap my brain around your very well done write up.
 
electrondady1: Apologies if it is not

Bruce S

[Adriaan Kragten]

A picture tells more than a hundred words but it takes some minutes to write some words and some hours to make a nice picture. But I hope the picture in my previous post supports the written explanation of the two different windings.

[Bruce S]

Yes they do.

Thanks for the extra work.

Bruce S

[Adriaan Kragten]

A picture of the rotating coil is added to my third post.

[Bruce S]

That one is a classic that truly helps understanding not only the 3-phase arrangement, it also shows the overlap of the coils and rectangle magnets, tho some of the mags look to be a wedge type .

Having the offset degrees is also a nice touch.

Cheers
Bruce S

[Adriaan Kragten]

There is a mistake in the formula for U given in the figure of the rotating coil given in my third post, as I forgot the radius r. I have added r in the figure and changed the formula. The modified figure is given as an attachment.

[ Specified attachment is not available ]

[Bruce S]

Adriaan;
Is the upper Ω=2π(<- this is the symbol for Pi) l(<---this looks to be a lower case "L" (rad/s) correct? I could not tell what the 3rd letter is .
BUT do know that unless the person is aware of using the "rad" button in their calculator for the (rad/s) their outcomes will not be correct.

Bruce S

[kitestrings]

The visuals are quite helpful to me, and your drawings are nicely done.  Thank you for taking the time Adriaan.

I also find some of the animated models can be helpful to put together things in motion, like:
https://www.animations.physics.unsw.edu.au/jw/electricmotors.html#alternator

[Adriaan Kragten]

Adriaan;
Is the upper Ω=2π(<- this is the symbol for Pi) l(<---this looks to be a lower case "L" (rad/s) correct? I could not tell what the 3rd letter is .
BUT do know that unless the person is aware of using the "rad" button in their calculator for the (rad/s) their outcomes will not be correct.

Bruce S

The f in the first formula is the frequency in Hz. A radian is the arc for which the arc length is equal to r. As 360° = 2 * pi rad, 1 rad = 360° / (2 * pi) = 57,296°.

The rotational speed Omega in rad/s can be transformed into the rotational speed n in rev/min by: n = 30 * Omega / pi rev/min (I can't write Greek letters on this forum).

The C in the second formula is a certain constant added to make the equation right. I don't know the value. The given coil has only one turn. If the coil has more than one turn, C decreases proportional to the number of turns per coil.

[SparWeb]

Possibly only academic interest, but when working with electric theory and mechanisms together, the capital Omega is too easy to mix up with the symbol for ohms.  Most people in physics abandon the capital Omega and use the lower-case omega for rotational speed.  The character looks like a rounded "w" and isn't used for other common terms in either mechanical or electrical physics, and now it is often used as a more unique symbol to represent rotational speed.  I still see the the capital Omega used in my aerodynamics textbooks for angular speed of propellers, but I steer clear of using it that way.

http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#avel



Since I'm looking at Hyperphysics...

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/motorac.html



The property that Adriaan is defining with the equations in his diagram "U" is often called "Electromotive Force (EMF)".  It refers to the implusion given to cause electrons to flow.  If the circuit is open, current cannot flow, and the voltage measured at the connections is equal to the EMF.  Once the circuit is closed, current flows, and the amount of current is defined by the EMF divided by the resistance.

[Adriaan Kragten]

Yes, it might be better to use the small omega in stead of the capital Omega. Some definitions of the symbols have been changed in time and sometimes I don't use the modern symbols. For the torque in Nm, I am still using Q but in recent books one is using T.

[SparWeb]

I still like Q for torque, especially in the electrical context,  because T=Tesla.

[Mary B]

Adriaan;
Is the upper Ω=2π(<- this is the symbol for Pi) l(<---this looks to be a lower case "L" (rad/s) correct? I could not tell what the 3rd letter is .
BUT do know that unless the person is aware of using the "rad" button in their calculator for the (rad/s) their outcomes will not be correct.

Bruce S

Program called mathtype, saves the result as a picture file you can post... there are probably freeware versions... I know word can do it but the version I have is really clunky to use

The f in the first formula is the frequency in Hz. A radian is the arc for which the arc length is equal to r. As 360° = 2 * pi rad, 1 rad = 360° / (2 * pi) = 57,296°.

The rotational speed Omega in rad/s can be transformed into the rotational speed n in rev/min by: n = 30 * Omega / pi rev/min (I can't write Greek letters on this forum).

The C in the second formula is a certain constant added to make the equation right. I don't know the value. The given coil has only one turn. If the coil has more than one turn, C decreases proportional to the number of turns per coil.

[Adriaan Kragten]

I still like Q for torque, especially in the electrical context,  because T=Tesla.

Our normal and Greek alphabet together simply have not enough letters to use for all possible units and dimensions. So duplication for different fields of knowledge is inevitable. In my free public report KD 35, I start with a definition of all units, dimensions and indices used in the wind turbine theory given in this report. I have used the same definitions as used in most books about the aerodynamics of wind turbines written in between about 1950 and 1990. But certainly some letters are used differently in other fields of knowledge and may be also in more modern books about wind turbine theory.

[MagnetJuice]

Adriaan, thank you for starting this topic.

Understanding how voltage is generated is very important for anyone that wants to build an alternator to make electricity from wind, hydro or human power.

Your explanation of how voltage is generated in a coil was very good. But like others have already said, it is hard to follow without having an image to visualize the process.

I took one of your images and colorized it and made the text easier to read. I also labeled the legs of the coils and added the direction of rotation of the magnets.



If I was a newbie and thinking of building an alternator, I would be looking at the 4 magnets, N1, S1, S3 and N3.
I would like to know how those magnets and coils interact, and how power is being generated in those 2 coils.

If it is not too much trouble, could you please elaborate on that?

Your explanation could be useful to a lot of Newbies, and old timers too.

As always, thank you for your contributions.

Ed

[Adriaan Kragten]

Nice modification of my lowest picture by adding colours. I will try to answer your question.

In my first post I give the second way of how a voltage is generated. This is when a straight wire is moving through a constant magnetic field. For a generator, the coil is static and the magnetic field is moving but the effect is the same. I assume that the magnetic field is almost constant in between two opposite magnets. As long as a straight leg of a coil is moving in between two magnets, a constant voltage will be generated. For the given position of my upper picture, the heart of the left legs of the coils U1 and U2 coincide with the heart of the north poles N1 and N3. The heart of the right legs of the coils U1 and U2 coincide with the heart of the south poles S1 and S3. So if the left leg of a coil is just opposite to a north pole, the right leg is just opposite to a south pole. This means that the direction of the voltage generated in both legs will be opposite but as both legs are connected in a loop, the voltages will strengthen each other.

After a certain angle of rotation, the legs come out of the magnetic field in between two opposite magnets and the generated voltage will be zero. After a certain larger angle of rotation, the left and the right legs of a coil come in the region of other magnets for which the direction of the magnetic field is opposite to the starting position. So now a voltage is generated which has the opposite direction as for the starting condition. So rectangular magnets and straight coil legs result in a block voltage which deviates strongly from a sinus. In reality the change of the magnetic field in the air gap is not as suddenly as described and for my lowest picture there is a small region for which only a part of the wire is in between two opposite magnets. The corners of the block shaped voltage will therefore be rounded and so the real voltage will look more like a sinus.

For an 8-pole generator, the situation is the same as for the drawn zero position after 90° rotation of the armature. So a rotational angle of 90° corresponds to a phase angle of 360°. So a rotational angle of 1° corresponds to a phase angle of 4°. In the figure it can be seen that there is an angle of 0° in between the north pole N1 and the left leg of the coil U1. The generated voltage is maximal for this position. So this position corresponds to a phase angle of 90°. In the figure it can be seen that there is an angle of 30° in between north pole N2 and the left leg of coil V1. So this corresponds to a phase angle of 90° + 4 * 30° = 210°. In the figure it can be seen that there is an angle of 60° in between the north pole N3 and the left leg of the coil W1. So this corresponds to a phase angle of 90° + 4 * 60° = 330°. So the difference in between the phase angles is 120° and therefore a 3-phase current will be generated in between the phases U, V and W.

[Adriaan Kragten]

The main content of my contribution to the discussion about this subject is presented in a new chapter 9: "Voltage generation in a coil" of my public report KD 341: "Development of the permanent magnet (PM) generators of the VIRYA windmills". This report can be copied for free from my website: www.kdwindturbines.nl at the menu KD-reports. The pictures are the same but are now given at a more logic place of the story.

[Adriaan Kragten]

In one of my earlier posts I have said that it might be possible to use coils which are shaped like the rotor of a Wankel motor. I have checked this and it is possible. This coil geometry is now also given in chapter 9 of KD 341 including some more back ground. I have added the picture of this coil geometry as attachment. I think that these kind of coils can be manufactured easier than coils with straight left and right legs.