Calculating magnetic field of a planar coil

Discussion in 'Physics' started by Rydog, Oct 14, 2014.

1. Rydog Thread Starter New Member

Sep 4, 2011
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I'm trying to model the magnetic field of a Tesla bifilar type coil (pancake coil). I've searched the web for similar problems but only found ones that were solenoidal, closed ring, toroidal, or other. This will be used for AC currents but right now would be happy with just an equation for z-distance above the center using a DC source. I know Biot-Savart's law comes into play, but the only example similar to mine that I found was for a ring. I had thought about just doing that for the amount of turns I have, each with a different radius and adding them together, but I would think some of the B field would cancel out. Any help is appreciated. Thanks.

2. MrAl Distinguished Member

Jun 17, 2014
2,551
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Hello there,

Well, your approximation with rings probably is not a bad one if the wire diameter is small compared to the diameter of the whole coil. The only thing different then between the multiple rings and the true spiral is that the rings change diameter gradually with the spiral so there is an averaging effect, but because if there is almost no difference the error will probably be smaller than the error seen in real life due to other influences anyway.

The difference in the complexity of the resulting equation however could be more drastic:with the multiple rings there is probably a closed form solution (because a single ring has a closed form if i remember right) but with the spiral it may be that the resulting equation has to be solved numerically. As the 'shape' gets weirder the equations get harder to solve in closed form unless the shape can be approximated with shapes having known closed form solutions. So the majority of the stranger shapes will have solutions that will have to be carried out numerically. If i remember right, even a simple elliptic ring has a solution which has to be carried out numerically, but even if the shape does have a closed form it may be very hard to find the reduction that leads to that simpler solution, so even then numerical calculations will eventually come into play.

If the diameter of the wire is not small compared to the diameter of the coil then the solution has to be derived from the actual shape because there will be significant differences. But then again this is true even for a single ring, where we usually take it for granted that the wire diameter is small compared to the ring diameter. If it is not, then even for a single ring we'd have to do multiple calculations by extending the number of physical dimensions to include the wire thickness, which means we'd be doing an infinite number of rings of different diameter and different Z placements in space, and then even the current distribution inside the wire might come into play.

Usually the multiple ring solution is used even for cylindrical coils because the result comes out "close enough".

This is an interesting question because we dont see 3d spirals that much in electronics, except in coils. I had a problem a long time ago where i wanted to relate the time display of a VCR tape (used to be used to record TV programs) on a VCR player/recorder. The tape is wound on a spiral shape inside the cassette package. What i found out was that the way the coil inside is formed plays a role in the accuracy of the time vs how much tape has been wound on the take up reel inside the cartridge. If it is wound tight then we get a slightly different result than if it is wound looser, and the tape thickness (a very small fraction of an inch) also varies. So the real world result would never match the exact equation but some error would have to be accepted. I found that it worked pretty good though, and i could tell WHERE the tape was to within about 1 minute or something for a 6 hour tape. It required taking two time measurements and then doing a little calculation.
This problem reminded me of that probably because the spirals of the tape and the wire are the same type and make the same shape. But one thing is clear about both: there will be real world errors due to variations that are hard to control.

Last edited: Oct 14, 2014

Nov 9, 2007
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4. MrAl Distinguished Member

Jun 17, 2014
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Hi,

That's a very interesting link, thanks for posting it.

The approximation they are using in Figure 3 is a good example of a coil i would NOT recommend approximating as circular rings (as they are doing) unless it is just for a study of how the system works approximately without the need for very exact calculations. It should show a result similar to an actual spiral calculation, but the approximations i was talking about would be more applicable to a coil that had fine wire and somewhat large coil diameter, and so there would be quite a few turns of wire involved, and they would be closely spaced like one right next to the other with only the enamel coating in between the two turns. I think it is somewhere around the twentieth turn that the difference in arc length between a spiral and circle starts to look insignificant (in contrast, for one turn only there is a pretty big difference), but again it really does depend on the accuracy you are looking for. In the real world a lot of times other variations contribute more to the error anyway

5. Rydog Thread Starter New Member

Sep 4, 2011
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Thank you both for your help! It was very useful.

6. MrAl Distinguished Member

Jun 17, 2014
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515
Hi,

You are welcome, and BTW the kind of spiral we are talking about here is the Archimedes Spiral.

Also interesting though is that the shape of a planar coil wound onto a coil form with a circular cross section is not a true spiral really because the start of the first layer creates an abrupt 'cliff' that the second layer has to climb to get up on top of that first layer. So when the second layer wire gets close to the start of the first layer wire it leaves the circular coil form as a tangent, which is a straight line, until it gets above the first layer and then the semi flexibility of the wire causes it to overshoot the top of the first layer and then eventually fall back down to actually meet the top of the first layer, which causes a 'bump' to appear where it meets the start of the first layer. So the shape is more like a combined spiral, straight line, and bump. The third layer will follow the bump a little better, but the bump will never completely disappear, so we get a very strange shape if we wanted to do it perfectly.
The bump will be small for smaller diameter wire.

A coil wound in free air could overcome this problem if it could actually be bent into a perfect spiral somehow that is