From: Paul
Date: Sat, 16 Feb 2002 12:21:05 +0000
Subject: [TSSP] Inductance of a flat spiral coil
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Hi All,
Just a bit of an interesting preprint...for your critical review,
John Tomacic wrote to me reminding me of a web page of Alan Sharp's,
and attached is my reply.
Regards All,
--
Paul Nicholson,
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Message-ID: <3C6E4CBF.8DFB19F2@abelian.demon.co.uk>
Date: Sat, 16 Feb 2002 12:12:47 +0000
From: Paul
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To: John Tomacic
Subject: Re: Flat secondary measurements
References:
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Hi John,
Grab some strong coffee, and fasten seat belt...
> http://ourworld.compuserve.com/homepages/alansharp/mathpage.htm
Yes, I've seen this before. Lets just check Alan's conclusion...
Alan writes:
> Lmax for a given length of wire
So we have a fixed wire length and a fixed wire diameter...
> ...for a variety of values of the radius
> The radius of the coil and the length of wire gives you the
> number of turns,
Ok, so we have turns = wirelen/coil_diameter/PI
> the number of turns multiplied by the wire diameter gives you the
> height of the coil.
So, height = turns * wirediam
Alan varied the radius, and derived everything else from that, I'll
instead vary the form factor (=height/coil_diam), so
turns = wirelen/coil_diam/PI
= wirelen/(height/form_factor)/PI
= wirelen * form_factor/(height * PI)
and since the height itself is turns * wirediam,
turns = wirelen * form_factor/(turns * wirediam * PI)
therefore
turns = sqrt( wirelen * form_factor/(wirediam * PI))
from which
radius = height/formfac/2
height = turns * wirediam
Note that my form factor is the more conventional H/D rather than
Alan's H/R, so we'll expect a max L at H/D = 0.45
A suitable acmi input file is attached. The results are
FORMFAC| TURNS| HEIGHT| CL.r| CL.h2| CL.n| CL.L| CL.R
0.100|116.546| 0.117| 0.583| 0.117| 116.5| 31.7 mH| 9.51
0.150|142.739| 0.143| 0.476| 0.143| 142.7| 34.0 mH| 9.51
0.200|164.821| 0.165| 0.412| 0.165| 164.8| 35.3 mH| 9.51
0.250|184.275| 0.184| 0.369| 0.184| 184.3| 36.0 mH| 9.51
0.300|201.864| 0.202| 0.336| 0.202| 201.9| 36.5 mH| 9.51
0.350|218.038| 0.218| 0.311| 0.218| 218.0| 36.7 mH| 9.51
0.400|233.092| 0.233| 0.291| 0.233| 233.1| 36.8 mH| 9.51
0.450|247.231| 0.247| 0.275| 0.247| 247.2| 36.8 mH| 9.51
0.500|260.605| 0.261| 0.261| 0.261| 260.6| 36.6 mH| 9.51
0.550|273.324| 0.273| 0.248| 0.273| 273.3| 36.5 mH| 9.51
0.600|285.478| 0.285| 0.238| 0.285| 285.5| 36.3 mH| 9.51
0.650|297.135| 0.297| 0.229| 0.297| 297.1| 36.0 mH| 9.51
0.700|308.352| 0.308| 0.220| 0.308| 308.4| 35.8 mH| 9.51
0.750|319.174| 0.319| 0.213| 0.319| 319.2| 35.5 mH| 9.51
0.800|329.642| 0.330| 0.206| 0.330| 329.6| 35.2 mH| 9.51
0.850|339.787| 0.340| 0.200| 0.340| 339.8| 34.9 mH| 9.51
0.900|349.638| 0.350| 0.194| 0.350| 349.6| 34.6 mH| 9.51
0.950|359.219| 0.359| 0.189| 0.359| 359.2| 34.3 mH| 9.51
1.000|368.551| 0.369| 0.184| 0.369| 368.6| 33.9 mH| 9.51
So it looks like Alan is correct on H/R = 0.9 as the best for max L
in a solenoid.
I've always assumed that a flat coil would do worse than a solenoid,
because the inner turns hardly have any area and therefore don't
contribute much to the L. Your observation yesterday made me sit up!
Just shows you can't take anything for granted!
Now lets take the same piece of wire above, and wind a range of flat
coils out of it...
Label the inner and outer radii R1 and R2, so that we can write down
wirelen = 2 * PI * turns * (R1+R2)/2
and
R2 = R1 + turns * wirediam
Lets define a 'spiral factor' as (R2-R1)/R2 to describe the shape of
the flat coil, so R1 = R2 * (1-spirfac). Then a spirfac of one gives
a spiral that closes in the middle and spirfac ranges between zero
and one.
Then
R2 = R2 * (1-spirfac) + turns * wirediam
so
R2 = turns * wirediam/spirfac
and from the first equation,
wirelen = PI * turns * (R1+R2)
= PI * turns * (2*R2 - turns * wirediam)
= PI * turns * (2*turns*wirediam/spirfac - turns * wirediam)
= PI * turns^2 * wirediam * (2/spirfac - 1)
So if we compute for a range of spirfac, the other quantities depend
as
turns = sqrt( wirelen/PI/wirediam/(2/spirfac - 1))
R2 = turns * wirediam/spirfac
R1 = turns * wirediam/spirfac * (1-spirfac)
Acmi input file attached.
The output is
SPIRFAC| TURNS| CL.r1| CL.r2| CL.n| CL.L| CL.R
0.050| 59.015| 1.121| 1.180| 59.0| 22.7 mH| 9.51
0.100| 84.551| 0.761| 0.846| 84.6| 27.5 mH| 9.51
0.150|104.944| 0.595| 0.700| 104.9| 30.4 mH| 9.51
0.200|122.850| 0.491| 0.614| 122.9| 32.3 mH| 9.51
0.250|139.299| 0.418| 0.557| 139.3| 33.6 mH| 9.51
0.300|154.822| 0.361| 0.516| 154.8| 34.7 mH| 9.51
0.350|169.742| 0.315| 0.485| 169.7| 35.4 mH| 9.51
0.400|184.275| 0.276| 0.461| 184.3| 35.9 mH| 9.51
0.450|198.581| 0.243| 0.441| 198.6| 36.2 mH| 9.51
0.500|212.783| 0.213| 0.426| 212.8| 36.4 mH| 9.51
0.550|226.984| 0.186| 0.413| 227.0| 36.5 mH| 9.51
0.600|241.273| 0.161| 0.402| 241.3| 36.6 mH| 9.51
0.650|255.733| 0.138| 0.393| 255.7| 36.4 mH| 9.51
0.700|270.442| 0.116| 0.386| 270.4| 36.3 mH| 9.51
0.750|285.478| 0.095| 0.381| 285.5| 36.0 mH| 9.51
0.800|300.920| 0.075| 0.376| 300.9| 35.7 mH| 9.51
0.850|316.853| 0.056| 0.373| 316.9| 35.4 mH| 9.51
0.900|333.367| 0.037| 0.370| 333.4| 35.1 mH| 9.51
0.950|350.562| 0.018| 0.369| 350.6| 34.9 mH| 9.51
which peaks at spirfac = 0.6, ie when the width of the winding
is 60% of the outer radius. The resulting L is slightly less than
the same wire wound into the optimum solenoid.
Suggests perhaps that the reason you got such a big gain in L
between your example solenoid and your flat coil, was that your
solenoid was far from its optimum L for the given wire
(formfac=21.9"/5.625"=3.9), whereas your flat spiral was closer
(spirfac=(11.1"-2")/11.1"=0.82) to its optimum of 0.6?
Maybe you'd like to confirm this conclusion, and if you agree with
it, inform the pupman list?
Best Regards
--
Paul Nicholson,
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; Sample input file for acmi.
;
; Finding the solenoid form factor which gives highest L for a given
; piece of wire.
;
; See
; http://ourworld.compuserve.com/homepages/alansharp/mathpage.htm
;
; This input file demonstrates Alan Sharp's conclusion that the max L for a
; given piece of wire occurs when H/D is around 0.45
;
; First the fixed quantities
;
wirelen = 1400 * 12/39.37 ; A fixed 1400 feet (in metres)
wirediam = 0.001 ; A fixed 1 mm diameter wire
;
; And we'll vary the form factor...
;
formfac = [from 0.1 to 1 step 0.05] ; Our desired range of form factors
;
; And now the dependant quantities...
;
turns = sqrt( wirelen * formfac/(wirediam * 3.14159))
height = turns * wirediam
;
; Finally, the coil description.
;
coil {
radius height/formfac/2
height1 0
height2 turns * wirediam
conductor wirediam/2
turns turns
}
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; Sample input file for acmi.
;
; Finding the flat coil dimensions which gives highest L for a given
; piece of wire.
;
;
; First the fixed quantities (same as sharp.in)
;
wirelen = 1400 * 12/39.37 ; A fixed 1400 feet (in metres)
wirediam = 0.001 ; A fixed 1 mm diameter wire
;
; Defining a 'spiral factor' to be (outer_radius-inner_radius)/outer_radius...
;
spirfac = [from 0.05 to 0.95 step 0.05] ; Our desired range of spiral factors
;
; And now the dependant quantities...
;
turns = sqrt( wirelen/3.14159/wirediam/(2/spirfac - 1))
;
; Finally, the coil description.
;
coil {
radius1 turns * wirediam/spirfac * (1-spirfac) ; Inner radius
radius2 turns * wirediam/spirfac ; Outer radius
height 0
conductor wirediam/2
turns turns
}
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Maintainer Paul Nicholson, paul@abelian.demon.co.uk.