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Re: [TCML] charging reactors - Inductance formulas: maximum values
Hi Bart, dean et al, etc
I was going to post some long discussion of more accurate inductance
formulas that apply over much wider ranges of size, dimension ratios etc
that what the traditional formulas do.
but I've realised the standard wheeler equations are good enough for
most applications as they only deviate by about 10% for most cases and
less than 1% for ratios of length to Diameter above 0.4
(hence the popularity of wheelers formula)
But this still assumes that the wire is small, number of turns is large,
turns are closely spaced and the wire/insulation size ratio is big.
Have a look at this page - (remember that most formulas are simply
rearrangements of wheelers formula which was infact a simplication and
approximation of Nagaokas more general formula.
http://www.g3ynh.info/zdocs/magnetics/part_1.html#wheeler
Hence all the common formulas take the form of R² x N² (Diameter) or
perhaps D² x N ² (radius)
which is then multiplied by some factor (depending on whether inches,
centimeters, or metres is used) , the divided by some other term
containing the length - and sometimes the length term is on top but n²
is used (turns per inch - instead of Total turns)
In simple terms the equations that use Nagaokas formula or Wheelers
approximation to his formula are:
Single layer coils: (measured in cm) L = PI² x (N²xD²) x K / length
Multilayer coils: (measured in cm) L = PI² x (N²xD²) x K' / length
all dimensions measured in centimeters (1 cm = 0.001 microhenries)
Where K is a correction factor which takes account of the ratio of
diameter to length
and K' is a factor which takes into account the ratio of length to
thickness as well as the ratio of diameter to thickness...(scroll down
its not that long a list)
example of K
d/l K
0.0 1.000
0.01 0.9958
0.1 0.9588
0.2 0.9201
0.3 .8838
0.4 0.8499
0.5 0.8181
1 0.6884
2 0.5255
3 0.4292
4 0.3654
5 0.3198
10 0.2033
20 0.1236
30 0.091
40 0.0728
50 0.0611
100 0.034
200 0.0199
300 0.01399
400 0.01095
500 0.00904
Sorry I lost the table in my book at home on K' - BUT I'm sure someone
can find it somewhere..
eg using these tables a 10 turn coil of almost infinitely thin wire that
is 1/PI diameter and a length 100 cm will give an inductance of 1 cm (in
other words it has a circumference of 1cm)
This is where the term cm comes from in the electromagnetic units that
Nikola tesla used to use to calculate inductance and capacitance in his
Colorado springs notes - it makes it more intuitive to calculate
inductances etc...(that is if you don't use inches everyday ;-) )
BTW a sphere of 1cm also has a capacitance of 1cm.
to optimize a coil for maximum inductance for a given size and wire
length (which also minimises the resistance) all the turns should be as
close as possible to each other so that the flux lines from each turn
cover the same wires (or as many as possible of the other turns) eg a
circle or a square,
and the radius should be as wide as possible - but as circles are hard
to make most people wind them in a square former - although you could
pack it as you wind it....
Basically the ideal shape is a toroid with the cross section and the
plan views both being a circle,
but most commercial inductors for sound systems crossovers are pretty
close to the ideal shape.
This shape is called the Brooks coil...
Brooks, who wrote a paper in 1931, calculated that the ideal value for
the MEAN RADIUS is very close to 3c/2
where c is the sides of a square that revolves around a radius - eg a torus.
in other words the middle of the square former is the radius 3c/2
|<c>| axis
| | | |
|< 3c/2 >|
The shape is not absolutely critical as a fairly large % deviation from
this shape will still have a large inductance....
But as the frequency increases this shape become less accurate as do the
following..
For high frequencies the coil which gives the best L/R ratio is one with
well spaced turns - but it is not possible in practice to predict the
best shape for any particular frequency.
Maxwell determined that if the section of a coil is square then the
maximum inductance is obtained when the diameter is 3.7 times the side
of the square.
Shawcross and Wells have given curves with the inductance obtained from
the same length of wire 1570.8m at 1mm diameter wire wound into
different shapes - they determined the best shape is one which has the
the diameter of the coil about 3 times the side of the square section of
the torus of the coil.
For the best for width(b) to depth (c) ratio - The square shape seems to
have the best inductance - this coil above gave 1.29 henries at max.
for ratio b/c = 2 Max Inductance = 1.24 Henries for ratio d/b = 2.3
For ratio b/c = 5 max inductance = 1.07 Henries for ratio d/b = 1.5.
The ABSOLUTE MAXIMUM Inductance from a given length of wire, the section
should be circular and the with a ratio of a = 2.575r where a is the
radius of the circular axis of the coil and r is the radius of the
cross-section - eg a torus...
The inductance is L = 5.35 * PI * N² * a * 10-³ microhenries
where N = total turns
For a SINGLE LAYER coil (ie classic tesla coil) where the turns are
close wound, the maximum inductance for a given wire length is:
diameter/ breath = 2.415
which is probably why tesla's magnifier was so fat and squat - maximum
inductance and hence maximum coupling and energy transfer to the
secondary before being fed to the extra coil...?
For a FLAT CIRCULAR COIL - the lowest resistance - and hence max time
constant/maximum inductance of a flat circular coil is is obtained when
inner radius r1 is about 0.4 of the outer radius r2.
This coil has 99% of the maximum efficiency when r1/r2 < 0.7.
I have not verified these sizes and shapes and equations personally -
but they should be easy enough to plot on a spreadsheet..
the best way to do this would be to have a constant turns per inch
ration and this simplifies the terms need to plot.
then the basic inductance equation adopts the form D²*Length with
everything else being constant.....
well I hope this helps out a bit with maximising inductances etc....
Regards
Stephen
bartb wrote:
Stephen Hiscock wrote:
<snip>
as you can see no-one seems to be consistent with terms they use, or
letters that assign to variables...eg why is "a" the radius?, why is
"b" the length?, they swap radius for diameter and vice versa for no
apparent reason and it doesn't make the formulas anymore accurate or
easy to use..
If we all started from first principles and derived the inductance
formulas from scratch there wouldn't be much of an issue - and I
believe Dr. Antonio Carlos M. de Queiroz has written a program to
derive the inductance for any coil....
Regards
Stephen
P.s I will post some more general formulas tomorrow on inductance and
calculating the Coil of Maximum Inductance - as I'm due to do some
work which I have to rush of to...
Hi Stephen,
Your correct, there is a lot of inconsistency out there and
inaccuracy. The formula I originally posted, I found again listed and
there is no mention to average radius's. As a matter of fact, it
distinctly calls out inner radius and outer radius (pictorially as
well) which for the particular formula, is absolutely wrong. For
example, take a look at this:
http://microblog.routed.net/wp-content/uploads/2008/10/pancakewheel.pdf
If you follow that paper, L will be roughly 1/2 of the real deal,
because it is a hack job of the Wheeler approximation.
What is correct and should be used here are Wheeler approximations
such as:
L(uH) = (0.8 * a^2 * n^2) / (6*a + 9*b + 10*c ) inputs in inches.
L(uH) = (31.6 * a^2 * n^2) / (6*a + 9*b + 10*c) inputs in meters.
Both assume:
a = average radius of coil
b = length of windings
c = difference between outer and inner radii
The single layer air core that you showed is also what the Proline
Calculator used (except they are calling out d as average diameter)
which is also a nice try but another example of a hack job.
L(uH) = (d2 * n2)/(18*d+40*l)
Regarding maxL. I look forward to seeing what you have there. Max L is
very useful in these type of multilayer inductor designs because as it
should minimize DCR and maximizes L for the least wire length (which
should be a goal with this type of design).
Take care,
Bart
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