The multi layer coil

Figure 1 below shows a multi layer air cored coil wound on a circular coil former or
bobbin.

This type of winding is very common because it's simple to construct with a winding
machine and a mandrel. We'll consider it in some detail. The first question is this: if
you have a fixed length of wire then what dimensions of the winding give the
greatestinductance? Put another way, what is the most efficient shape?
The ratio of the winding depth to length, which is c/b, needs to be close to unity; so
the winding should have a square cross section. This makes sense because only with
the square is the average distance between turns at a minimum (a circular cross
section would be even better, but that is hard to construct). Keeping the turns close
together maintains a high level of magnetic coupling ('flux linkages') between them,
and so the general rule that the inductance of a coil increases with the square of the
number of turns is maintained.

OK, now what about the mean radius of the turns: dimension a ?
In the adjacent Fig. 2 you see three coils in cross section. Each uses the same length of
wire but the diameter of the coil varies. The inductance of any one turn is linearly
proportional to its diameter; so you want a large diameter to get the most inductance.
Also you need all turns in the winding to be as close as possible to all the others. The
coil on the left fulfills these requirements, but it has a problem because, in making the
diameter large, you don't have sufficient wire to give it many turns. Since the
inductance of the winding as a whole varies as the square of the number of turns the
left hand coil won't have high L.
The coil on the right does have a high number of turns, but it isn't optimum for two
reasons. The diameter of each turn is small (particularly those near the centre of the
coil) which leads to a low inductance per turn. Worse, the distance between turns
separated by the diagonals of the winding cross section is large. This leads to weak
coupling between them (lower flux linkage) and a failure of the N
2
relationship with
coil inductance.
Can you anticipate a punch line? Brooks, who wrote a paper in 1931, calculated that
the ideal value for the mean radius is very close to 3c/2. We call a coil having these
dimensions a Brooks coil. It's worth emphasising that the Brooks ratio is not critical.
You can have a coil which deviates from it quite significantly before L falls off very
much. Also, you may have other considerations than the inductance alone.
If we let S
1
= (c/2a)
2
then the inductance, in henrys, of any of our three coils can be
approximated by
L = 410
-7
a N
2
((0.5+S
1
/12)ln(8/S
1
) -0.84834+0.2041S
1
)
Equation ACA

where a is the mean radius of the winding in metres.
Example: If the mean radius is 2.4 cm, the width is 9mm and N is 350 then L = 6.977
mH.
If we have a proper Brooks coil then the formula above boils down to
L = 1.699410
-6
aN
2
henrys
Equation ACB

Example: If the mean radius is 2.4 cm and N is 350 then L = 4.996 mH.
What is an air core coil?
We'll use the term 'air core coil' to describe an inductor that does not depend upon
a ferromagnetic material to achieve its specified inductance. This covers the cases
where there really is just air inside as well as windings upon a different insulator such
as bakelite, glass or PTFE etc.
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Do you need an air coil?
What are the 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 converters. In circuits such as
audio cross over networks in hi-fi speaker systems you must avoid distortion;
then you need an air coil. Most radio transmitters rely on air coils to prevent
the production of harmonics.
Air coils are also free of the 'iron losses' which affect 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 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. Balanis has the gory details. 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 field may be what you want.