Calin Dorohoi wrote...
I would like to know how to compute the shortest wire length
required to build a specific inductance given a certain wire
diameter. My question refers especially to multi-layer
air-core coils.
This is a problem that out distant forefathers solved long ago.
The optimum highest inductance to wire-length ratio occurs in the
Brooks inductor. Its windings are square in cross section and have
a roughly 3:1 ratio of mean-diameter to length. Here's an excerpt
from a post I made on Dec 28th 1977, commenting on an article by
Robert Kesler, "Multilayer Air-Cored Coils," Electronics World +
Wireless World, Sept 1997 pg. 752-753.
"According to Grover, the theoretical optimum ratio for a coil with
minimum wire length, using a square-cross-section, has a mean-dia/
length ratio = 2.967. The famous 'Brooks Inductor' closely matches
this with an easy-to-remember 3.00. Your coils aren't quite square,
if I understand correctly, and are the equivalent of about 3.18.
"In Prof. P.N. Murgatroyd's definitive paper on this type of coil,
'The Brooks inductor: a study of optimal solenoid cross-sections,'
in IEE Proc., vol 133 B, 5, pp 309 - 314, Sept. 1986, we find that
(hard to wind) circular cross-section coils are about 1.1% higher
inductance than square coils, and rectangular coils are somewhat
worse. We also find that the ideal ratio 2.967 is at the center
of a rather broad minimum.
"One other comment. Brooks coils are non-optimum for many purposes.
Used at high-power, they have nasty overheating problems in the wire-
packed center. Murgatroyd has some nice design curves for picking
wire sizes and coil dimensions, based on wire current density. Other
coil configurations have much better air- or fluid-cooling access.
"Brooks coils, if used at moderately high frequencies, have very high
ac proxmity-effect losses (even when made with litz wire, I found),
but this can be dramatically reduced with other coil configurations."
I wrote about the other configurations at various times back then.
------
This was a busy time. This is from another post I made the same day:
* In Grover's famous book "Inductance Calculations" (Van Nostrand 1946,
reprinted Dover 1962), page 95, we find an exact formula for solenoids
wound with square cross-sections,
(2) L = 0.001 a N^2 Po uH/cm.
Po is a function of c/2a and is given in a lengthy table, varying from
60 for very short coils to a maximum of 7. For small cross-sections he
shows a hairy formula credited to Stefan in 1884.
* Relevant to this is the Brooks coil, a square-cross-section solenoid
that is an air-core-inductor champion: requiring minimum wire for a
given inductance. The cross-section below shows the Brooks coil's
easy-to-remember "four-squares" geometry: Two wire squares, spaced two
squares apart.
|
##### | #####
##### | #####
##### | #####
|-- a --|
| coil axis
The coil's mean diameter is 3.00 times its length and the winding height
equals its length. Grover (pages 94 and 98) provides a simple, precise
formula for the theoretical inductance. Rearranged, we have
(3) L = 1.35234 mu_o a N^2
where mu_o is the permeability of free space = 4 pi 10^-7 H/m
and a is the coil's mean radius.
(4) L = k1 Do N^2 N = sqrt ( L / k1 Do)
where Do is the outer diameter (twice the coil-form diameter),
and k1 = 0.006373 uH/cm or 0.016187 uH/inch.
--------------------
Here is what Bob Kesler wrote on Dec 27th (I have edited slightly):
The classic is the Wheeler formula. It has been around for some
70 years. I quote its metric version:
L = (7.87*N^2*M^2)/(3*M+9*B+10*C)
L inductance, nanoHenryes
N number of turns
M mean diameter of the coil
B width/length of the coil
C radial thickness of the coil
all dimensions in millimeters.
The formula is within +/- 1% if the three members of the denominator
are about equal. The above formula is only for calculating the
inductance, if you already know everything else. If you want to
_DESIGN_ a coil, use my method: It is for calculating the dimensions
of the coil, the # of turns AND the WIRE DIA from the given INDUCTANCE
and RESISTANCE of the coil. It yields the coil using the least amount
of copper possible.
In a nutshell:
0) Define target inductance (L nH) and resistance (R Ohms)
1) Calculate ideal mean dia of the coil: Mi = 0.0354*sqr(L/R)
2) Calculate the ideal # of turns: Ni = 1.07*sqr(L/M)
3) Calculate the ideal wire dia: Wi = 0.253*M/sqrN (mm)
4) Choose a realistic, available wire dia: W >= Wi
5) If the real Wire dia is different then Wi calculate M again:
- M = 3.08*L^0.2*W^0.8
6) The # of turns: N = 0.61*(L/W)^0.4
7) Coil length: B = M/3
8) Coil inner dia: I = 0.7*M
9) Coil outer dia: O = 1.3*M
10) The radial thickness of coil: C = 0.3*M
11) Check the resistance of the coil: R = M*N/(14250*W*W)
I doubt it's worth the time to
