planar coil magnetic field strength calculation

Hello, is there any calculator or formula that is straight forward where I could calculate the B field strength adjacent aka right next to the flat plane of a planar coil?

Also would that formula take into account frequency/reactance of the coil and B field strength within given parameters because I would like to use the coil as an air core coil.
thanks.

The field strength in the axis of a rotational symmetrical coil can be easily calculated by applying Biot-savart's law.

Well I looked at the many variants of the Biot-savart and I have to say my math is incapable of applying it to my case, can you please help?
I found the formula for a single circular loop but my case is a planar coil with multiple turns and each turn as it spreads out has a different diameter so I can't understand how can I calculate it.

You'll sum up the contributions of all turns. Complex cases beyond a multiple circular turns arrangement can't be solved by a simple formula. There's neither an analytical solution for the inductance. You need to refer to numerical 2- and 3D solvers.

so i can sum up the turns even though each next turn is larger than the previous one in its diameter?


Is the field at least close to uniform and homogeneous in each of the two flat planes of the coil? In a sense that the field lines travel at near 90 degrees perpendicular outside one of the planes loop around and come back through the other one ?


Maybe you have the formula at hand? the ones i see online tend to differ

so i can sum up the turns even though each next turn is larger than the previous one in its diameter?

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No, every turn has different r and a according to Biot-Savart and contributes different H amount.

Salvador12 said:

Is the field at least close to uniform and homogeneous in each of the two flat planes of the coil? In a sense that the field lines travel at near 90 degrees perpendicular outside one of the planes loop around and come back through the other one ?

Click to expand...

This is an old picture that I created for wireless charging, showing the H field. Arrow size indicates magnitude.

Salvador12 said:

Maybe you have the formula at hand? the ones i see online tend to differ

Click to expand...

Not sure how you intended on doing this calculation, but if its matlab, here is some handy code made at our work long ago that can calculate the B field of 2 coils at any point in space. This example is for a Helmholtz configuration, so you could presumably delete the part of the code dealing with the second coil, make this code a function with radius as an input, and sum up the contributions of a 1 loop at a time. I did something very similar for our coils, and got reasonably good accuracy.

Hope it helps,
Sami
View attachment ARL coil document.pdf
(Code is near the end of the document, and its a bummer the last page is missing)

Ok, thanks folks for the input , I appreciate , now I see that to calculate the field for such a planar multiturn coil is really a headache.
To be honest to know whether my experiment would work at all I only need to know the approximate B field not precisely down to decimal digits.
0.5 Tesla would give me a start , 1T would be perfect, what do you think can I get something like these field strengths out from an air core (because high frequency) planar coil say made from a copper tube with some 8/10 turns ? Calculating in a solenoid calculator tool I get I would need some 1000 to 2000amps of current and interestingly enough it shows negative inductance for the coil which is either a glitch in the calculator due to my crazy inputs or may be a thing for such a coil?


thanks.

Salvador12 said:

0.5 Tesla would give me a start , 1T would be perfect
thanks.

Click to expand...

My intuition says, that's not happening. You do realize how strong of a magnetic field that is? That's close to MRI and Hall measurement field strengths (both tools require cooling in order to function).
I can work that code I posted if you want to provide some better details on your structure. Wire diameter, coil pitch... etc... but you wont like the answer. I would be surprised if you get more than a few 100 gauss (0.01 T) even with 1 amp worth of current. But, we might as well see what the math says.
Sorry,
Sami

Well 1 Tesla is not very strong for a transformer core or a inductor ,but it is strong for a air core coil which is what you meant right?

I would be very thankful if you could help me out with the code,
well for some general parameters I would give these.
A planar coil, made from a hollow copper tube where tube diameter is 1cm. The coil outer diameter would be 26 cm and inner diameter 5cm (as in the first turn starts on a circular line which has 5cm diameter because in the very middle there is an axis)
the coil would have about 15 to 20 turns depending on the free space etc.
I could supply the coil with very high current if neccesary but I would need to know the inductance of the coil first, so for current please assume any current that would be neccesary to build a 1 Tesla field.

I wonder what kind of results the code will show.

You mentioned by the way that you want to generate a field at "high frequency". How high?

Basically RF range, everything from about 1Mhz up to how high I can get with a coil that has certain physical properties.
For lower than 1 Mhz frequencies I wouldn't have a problem creating a strong B field up to 1 Tesla because I could use ferrite cores, so I am looking into the air core coil situation to know how strong of a field I could get out of it which would then determine my further actions.

This code breaks a planar coil down to the ideal case. The coil is broken down into individual current loops we apply biot savart to. I can only provide the field strength, I cannot comment on the inductance of the structure.

However, a quick google search yields above. I can only assume that’s uH since all other units are micro.

I’ll start with some numbers for planar coils I’ve used in the past. The much smaller dimensions will produce more intuitive results, best to start there. This coil has an inner radius of 50um, a pitch of 20um, and for this example I used 1Amp

The first plot shows the field strength as a function of position in space with respect to the coil. 0,0 is the center of the coil. You can see the field contributions of each loop. The second plot is the field strength at Z=0 along the x-axis. You wanted to know the field strength adjacent to the coil. In this example the last coil resides at x=230um (0.81 T). 20um away, at x=250um, the field strength is 0.0165T. This will directly scale with current. If we need 1T at 250um, we would need 100A.

Now lets look at your coil. Inner radius is 2.5cm. Since your wire diameter is 1cm, I choose a pitch of 1.1cm to cram in as many turns as possible. With 10 turns you get an outer radius of 13.5 (bit too large, close enough). I bumped up the current to 10A so the numbers aren’t too small.

If you look closely you can see the contributions of the individual loops in the first plot. The second plot shows the field strength as a function of distance. The last loop is at x=124cm and ~2mm away the field strength is ~.001T. Again, that will scale with current, to get 1T you would need 10,000 Amps.

Take it all with a mountain of salt, but i dont think you will ever get field strengths that large with a planar coil (considering µ_r=1). I can send you the code if you want to double check these numbers.

Well from an intuitive viewpoint 10K amps sounds an awful lot to get a 1T field in a coil but I guess thats the way it could be.

Well since for high frequency I can't use any materials that would help with field strength then I ave to ask what would be the best air cored coil to use in order to achieve a decent field strength on a circular flat surface of various sizes?

Salvador12 said:

Well from an intuitive viewpoint 10K amps sounds an awful lot to get a 1T field in a coil but I guess thats the way it could be.

Click to expand...

Skin effect will be an issue. MRT with 1T field strength require cooled superconducting coils to achieve such field strength, at somewhat larger coil sizes.

I dont know what your application is, but if you ever create a 1GHz 1T unit that doesn't require liquid helium, you will become a billionaire overnight, and people like GE will be knocking on your door to scoop up your patent.

another consideration: 1mH inductor at 1 GHz: 2*pi*1GHz*1mH = 6.2 MOhm. Props to google btw, can't believe that worked and the units were maintained.

Yes I do realize the task is near impossible but on the brightside and MRI unit is large and the field needs to be strong in a large area , for me the field or I should say the field lines dont have to travel long , my field gap is much smaller.
And I only need the required field strengths some few cm right next to the coil.

Well sure a 1mH inductance is no good at those frequencies, would need a lightning strike to drive such a coil.
But since I need the field on a flat surface I was thinking could I make a weird looking LC circuit by making two parallel plates to form a capacitor (dielectric in between) and then on the rear surface of one of the plates form the whole surface area with thick little multiturn coil or somehow stack them and connect them all in parallel.
As the capacitor would charge/discharge the current would run through those coils , if the current would be large enough (coil inductance very low) then maybe I could get a field strong enough?

So what is your requirement/spec for operating frequency?

well ideally as wide as possible, I am trying to make a sort of a frequency generator , for lower frequencies getting the field strength is not a problem but for higher ones it becomes a problem, so here i'm looking for everything that is above 1Mhz up to as high as physically possible because eventually everything will have its constraints