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The formula used in this L (uH) =31.6*N^2* r1^2 / 6*r1+ 9*L + 10*(r2r1) where…

NOTEThis formula applies at ‘low’ frequencies (<3MHz) using
enameled copper wire tightly wound.
Please note that the diameter is measured from center of wire trough
center of the coil and to center of the wire on the opposite side.
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The calculation is based on Wheeler’s 1928 formula for a singlelayer solenoid which is given in its original form as:
L = a² N² / (9a + 10b) [microHenries] , b > 0.8a
Where b is the coil length in inches, and a is the radius in inches.
To convert this formula to SI units, we will use the symbols r = radius, D = 2r = diameter, l = solenoid length.
Factoring b from the denominator gives:
L = 106 a² N² / [ b (10 + 9a/b)] [Henrys]
The quantity a/b is dimensionless, and so we can immediately substitute in the denominator:
L = 106 a² N² / [ b (10 + 9r/l)] = 106 a² N² / [ b (10 + 4.5 D/l)]
Factoring 10 from the denominator gives:
L = 107 N² ( a² / b ) / (1 + 0.45 D/l) [Henrys]
where..
noteThis formula applies at ‘low’ frequencies (<3MHz) using enameled copper wire (magnet wire) close wound.
Tip 1Small reductions in the inductance obtained can be achieved by pulling the turns apart slightly. This will also reduce selfresonance. Other combinations of wire and coil diameter may be tried but best results are usually obtained when the length of the coil is the same as its diameter.
Tip 2 If you need good induction stability in the presence of vibration then wind the coil on a support made from a suitable non magnetic plastic or ceramic former and lock the windings using epoxy glue or other suitable adhesive.
A flat spiral coil is a type of an air core inductor
usually incorporated in the primary of a tesla generator,
RFID tag, and proximity detectors. In the same category
as the flat spiral coils we have planar spiral coils,
planar square spiral coils, planar rectangular spiral
coils, planar hexagonal spiral coils and octagonal spiral
coil. Planar coils are mostly used in high frequency
applications and designed as tracks on a circuit board.
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, platsic, ceramic etc.
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 nonlinearity in the magnetization curve can be tolerated; for example in switching power supplies and in some switching topologies this is an advantage.
In circuits such as audio cross over filters in hifi 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 Qfactor, 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.
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 selfresonance dur 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.
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