air core coil equations

Discussion in 'General Electronics Chat' started by aidafiza, Jul 8, 2008.

Jun 2, 2008
17
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hi guys..

Is there any other equations to calculate inductance for a single layer air wound/core coil inductors other than the Wheeler's formula? I have made some air wound coils inductors on my own and checked the inductance using an RLC meter and compared the value with the calculated value using wheeler's formula. the values i got are very different from calculation....i also use the calculator in this website:-
http://www.vk2zay.net/calculators/
it provided me two answers;one using wheeler's formula and another one is a formula i don't know of..it says "Long, no fringe correction" in the website..does anyone know what formula the website use??

(i have attached the wheeler's formula with this post..)

• wheeler.jpg
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2. SgtWookie Expert

Jul 17, 2007
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Wheeler's Formula (single layer air core coils)
L(uH) = (r^2) * (N^2) / (9*r + 10*h)
where:
N = number of turns
h = coil height (or length) in inches

Jun 2, 2008
17
1
that's the formula I have attached in my previous post but the one i attached uses centimeter for measurements instead of inches..the details for the equation I've attached is as follows:
L is the inductance in microhenry
r is the radius of the coil in centimeter
N is the number of turns of the coil
A is the length of winding in centimeter

but thanks anyway sarge

4. The Electrician AAC Fanatic!

Oct 9, 2007
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Jun 2, 2008
17
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thanks..the answer i got is still the same.the program you suggested uses wheeler's formula too..
the "Long, no fringe correction" answer from the website i mentioned gave more accurate answers but i don't know what formula/equation the author of the website uses. any idea??

6. The Electrician AAC Fanatic!

Oct 9, 2007
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Please post the parameters of your coils and the calculated inductances you get from various formulas.

7. The Electrician AAC Fanatic!

Oct 9, 2007
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The formulas used by the program at this site:

http://www.moshier.net/coildoc.html

don't use Wheeler's formula.

"This program computes the electrical self inductance of various shapes and sizes of air core coils, using accurate formulas from National Bureau of Standards circulars and related sources."

Jun 2, 2008
17
1
I've sent an email to the calculator websites' owner and this is his reply:-

"The other result is for an "long" single-layer solenoid, and ignores the
fringing fields that result from its finite length. The fringe field's
flux leakage means it doesn't cut all the turns so the self-inductance
measured is lower than calculated using this model. You might think of
it as an upper-bound on the inductance you are likely to get.

It is calculated using:

L = (u * n^2 * A) / l

Where:
u is the permeability of the core (k * u0 or free-space u0)
n is the turn count
A is the cross-sectional area of the coil (m^2)
l is the length of the coil (m)

u0 = 4*pi*1e-7 H/m

For small coils your measured values may be larger than the Wheeler
calculated value because of the thickness of the wire. The diameter
should actually be the average "current sheet" diameter, so you might
get a closer result if you add the wire thickness to the diameter (which
will increase the inductance calculated a bit).

The Wheeler formula is normally quite good, accurate to a few percent
for the average single-layer solenoid you'd use in RF applications. It
is specified as being accurate when l > 0.4 * d:

L = (d^2 * n^2) / (l + (0.45 * d))

Where L is in uH and the l and d in metres.

For shorter coils there is a series of length to diameter tables
these days I'll improve the coil calulator with other formula."
-Mr Alan Yates-

So,thanks to Mr. Yates for the equations
and thanks to you guys as well..you've been great help..

9. SgtWookie Expert

Jul 17, 2007
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1,728
I've been playing around with the attached "InductorCalculator2" Excel spreadsheet for a while. Don't happen to have a good way to measure inductance at the moment, so I don't have a way to verify it's output.

It has two separate formulas; one for multilayer air coils and Wheeler's Formula. It figures in the diameter of the wire gauge that you choose, and you can also allow for the thickness of the insulation.

Feel free to play with it. It uses table lookups which may be confusing if you haven't done many Excel spreadsheets.

I'd like to hear how it's output compares with what you're reading from your wound inductors.

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10. The Electrician AAC Fanatic!

Oct 9, 2007
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The well-known version of Wheeler's formula was published in 1928. He published an improved formula in 1982 which isn't so well known.

It is possible to tweak his formulas for about 5 times better accuracy, and after applying the correction for round wire found in the Grover book, one can obtain a formula accurate to about .02 % for almost all coils, long and short.

See the attached .gif file

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Jun 2, 2008
17
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I think one of the reason for my measured and calculated inductance values to be vastly inaccurate are because of the windings are not constant, meaning some are winded loosely and some very tight. This is my first time winding my own inductors and I did not use specific winding equipments to make them.

Another reason could be the error and inaccuracy in measurement. Since the inductors are too small (2 to 6 milimeter long and 4mm in diameter) it was a bit hard to get the readings using the RLC meter I have. And the measuring gauge..well,I just used a plain ruler to measure the length and diameter..Optical errors are inevitable....

I guess skills and experience (and hi-tech,expensive equipments) are needed to wind a better inductor.

12. The Electrician AAC Fanatic!

Oct 9, 2007
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You are quite right. Here is a picture of an inductor constructed in such a way as to have a calculated value very close to its measured value.

The form is a ceramic cylinder, ground to precisely .75 inches in diameter. There is a groove ground in the cylinder so that the wire is held in place, and the groove depth is such that the diameter of the winding as measured from center to center of the wire is also .75 inch. The wire is 29 gauge which gives a wire diameter of .112576 inches. The length of the winding is exactly 4.5 inches and there are 180 turns.

If you plug these numbers into the 1982 Wheeler formula (with correction for the round wire), you get 93.5 uH. The inductance measured on a Wayne-Kerr precision impedance analyzer is 93.3 uH, which is about .2% different than the calculated value.

The 1982 Wheeler formula is good to about .02%; the Wayne-Kerr isn't that good. You can see the difficulty in verifying the claimed .02% accuracy of the formula!

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