# [TCML] charging reactors - Inductance formulas

Stephen Hiscock stephenhiscock at yahoo.com
Sun Mar 8 00:15:24 MST 2009

```After I posted a different Formula a day or so ago people have obviously
noticed that there are several different LOOKING formulas - which if
analyzed work out to be approximately the same.
The answer to this is obvious - each formula is either ONLY an
approximation for different inductances for the most common situations
or it is simply just a rearrangement of the equation, or a substitution
from metric to imperial, or simply cancellation of terms that become
very small when other factors are very large to enable a more
"Simplified" equation for practical purposes.

eg these things all affect inductance to small or BIG degrees depending
on the situation...
Shape (eg barrel, tubular, square, flat pancake, turns SPACING, ratios
(length, width, etc), THE FREQUENCY that the wire will carry (most
formulas assume HIGH frequency - because inductance is easier to measure
at high frequency), The NATURE of the CONDUCTOR ( eg iron is different
than copper), wire size, wire SHAPE (yes this affects it a LOT
-depending on different factors infact most formulas actually ASSUME
FLAT TAPE WIRE - NOT circular wire....- because it is easier to
initially generate the formula..

"A little knowledge is a dangerous thing" is a saying that has relevance
to this field of calculating inductances of coils.
eg if the "coil" is very long (1 mile long?) and only has 2 or 3 turns,
and a diameter of 1 inch, then for all practical purposes the "COIL"
will behave just like a straight length of wire.....

Also many formulas which look different are in fact exactly the same...

The formula I wrote the other day just to show that 1 henry is not an
unusual value is usually used for a single layer air core - but it is a
reasonable one for most other values especially when the number of turns
is very high and the aspect ratio is kept "reasonable"

L= (d² * n²)/(18d+40*l) (MicroHenries
where:
L is inductance in micro Henrys,
d is coil diameter in inches,
l  is coil length in inches, and
n is number of turns.

So in answer to Deano (question posed on 8/03/2009 at 12.48pm)
This is EXACTLY the same as the equation
L = (r²*n²)/(9r + 10H)
where H = length, d=2r (r=radius)
if you substitute d for 2r you will see the 4 on the top cancels and
leaves the other....

Now regarding formulas specifically designed for multilayer coils
one was posted by Dean and jim lux from the AARL handbook: (commonly
called Wheelers as the above are too)
L (uH) = 0.8 * a2 * n2 / (6*a + 9*b + 10*c )
where
a = average radius of windings
b = length of the coil
c = difference between the outer and inner radii of the coil.
all dimensions in inches.

Another by Bartb (6/4/2009 at 3.59pm)
(31.6 * n² * r²) / (6* r + 9*L + 10*(r0-r1))
n = total turns
ri = inside coil radius
ro = outside coil radius
L = coil length
Now this formula is CLOSE to the above, but not exact as someone along
the way has done a conversion from metric to inches and not quite got it
right - but it agrees within 0.1 or 0.3%
that number in front should be 31.49 - but whos' counting????
It is no different from the others above
(as long as people use "reasonable" numbers and ratios etc)
all dimensions in metres.

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...

bartb wrote:
> Hi Kurt and All (thanks for all your replies),
>
> I looked at all the formula's given and they all appear to be Wheeler
> from an air core inductance derivation. The formula that I used, I'm
> not sure of it's nature, but it is far off. However, the proline
> calculator is still significantly off as well from these Wheeler
> formula's by about 1H for the coil example I used. In my spreadsheet,
> layers are an input and used to determine total turns. Although the
> calculators turns and layers match my own, the Wheeler formula shows
> 3.95H where proline shows 5H. So, there is something else proline is
> looking at. However, the Wheeler formula (however you use it), is an
> air core multilayer coil formula.
>
> A magnetic mass such as ferrite, iron, or various composites will vary
> permeability substantially. While researching this aspect, the
> following formula was predominant:
> L=(0.4*pi*µ*N^2 *A*10^-8 )//l/
> where:
> L=inductance (henries)
> µ=core permeability
> N=number of turns
> A=core across section (cm^2 )
> /l/=magnetic path length(cm)
>
> The permeability of cores can vary widely, so one has their work cut
> out for them using a magnetic core.
>
> But as Kurt mentioned, maybe an air core inductor would suffice. This
> would certainly make the coil design far easier and L far more
> manageable. Considering the low current, it just might be the way to
> do it.
>
> Take care,
> Bart
>
>
> Kurt Schraner wrote:
>> Hi,
>>
>> Bart, I apparently seem using the same formula as the guy on
>> http://www.pronine.ca/multind.htm
>>
>> The formula seems to stem from the radio amateurs handbook. I took
>> mine from:
>>
>> ARRL Handbook 1957 / p27
>>
>> a=avg.diameter in inches = (Di+Da)/2
>> b=length of winding in inches
>> c=radial depth of winding in inches = (Da-Di)/2
>>
>> L[uH]=(0.2*a^2*n^2)/(3*a+9*b+10*c)
>>
>> I've verified it's results on my induction coils (without core, of
>> course), and it works well, i.e.:
>> My 26k turns induction coil secondary: measured 15.15H; calc. 15.09H;
>> Error 0.42%
>> My 48k turns induction coil secondary: measured 49.8  H; calc. 47.7
>> H; Error 4.22%
>>
>> More precise methods could probably been taken from NBS circular 74,
>> but the above is about within the precision of my LCR meter.
>>
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