Formula used in this calculation is from Wheelers approximations
which is accurate to <1% if the cross section is
near square shaped.:
L (uH) =31.6*N^2* r1^2 / 6*r1+ 9*L + 10*(r2-r1)
L(uH)= Inductance in microHenries
N = Total Number of turns
r1 = Radius of the inside of the coil in meters
r2 = Radius of the outside of the coil in meters
L = Length of the coil in meters
This formula applies at 'low' frequencies (<3MHz)
using enameled copper wire tightly wound.
More about air core inductors
What is an air core inductor?
An "air core inductor" is an inductor that
does not depend upon a ferromagnetic material to achieve
its specified inductance. Some inductors are wound without
a bobbin and just air as the core. Some others are wound
on a bobbin made of bakelite, plastic, ceramic etc.
Advantages of an air core coil:
Its inductance is unaffected by the current it carries.
This contrasts with the situation with coils using ferromagnetic
cores whose inductance tends to reach a peak at moderate
field strengths before dropping towards zero as saturation
approaches. Sometimes non-linearity in the magnetization
curve can be tolerated; for example in switching power
supplies and in some switching topologies this is an
In circuits such as audio cross over filters in hi-fi
speaker systems you must avoid distortion; then an air
coil is a good choice. Most radio transmitters rely
on air coils to prevent the production of harmonics.
Air coils are also free of the "iron losses"
which a problem with ferromagnetic cores. As frequency
is increased this advantage becomes progressively more
important. You obtain better Q-factor, greater efficiency,
greater power handling, and less distortion.
Lastly, air coils can be designed to perform at frequencies
as high as 1 GHz. Most ferromagnetic cores tend to be
rather lossy above 100 MHz.
And the "downside":
Without a high permeability core you must have more
and/or larger turns to achieve a given inductance value.
More turns means larger coils, lower self-resonance
due to higher interwinding capacitance and higher copper
loss. At higher frequencies you generally don't need
high inductance, so this is then less of a problem.
Greater stray field radiation and pickup:
With the closed magnetic paths used in cored inductors
radiation is much less serious. As the diameter increases
towards a wavelength (lambda = c / f), loss due to electromagnetic
radiation will become significant. You may be able to
reduce this problem by enclosing the coil in a screen,
or by mounting it at right angles to other coils it
may be coupling with.
You may be using an air cored coil not because you require
a circuit element with a specific inductance per se
but because your coil is used as a proximity sensor,
loop antenna, induction heater, Tesla coil, electromagnet,
magnetometer head or deflection yoke etc. Then an external
radiated field may be what you want.
An interesting problem is to find the maximum inductance
with a given length of wire. Brooks, who wrote a paper
in 1931, calculated that the ideal value for the mean
radius is very close to 3A/2. As can be seen from the
picture below, the coil has a square cross section (A=B)
and the inner diameter is equal to twice the height
(or width) of the coil winding.
We call a coil having these dimensions a Brooks coil.
Brooks ratio is not critical. You can have a coil which
deviates from it quite significantly before the inductance
falls off too much. Also, you may have other considerations
than the inductance alone.
The inductance for a Brooks coil can be found from
the following equation:
where A is the height and width of the coil winding
(in cm) and N is the number of turns. A second formula
is shown below:
where r is the mean radius of the inductor (in cm)
and N is the number of turns.
(r=mean length of the coil radius measured from the
center of the coil to the center of the coil height,
as shown in the figure above.)