where L is the inductance in henrys, N is the number of
turns, µ0 is the permeability of free space (4π x 10− 7 H/m),
A is the cross-sectional area of the coil in square meters,
and ℓ is the length of the coil in meters. This formula
applies only to long single layer coils, but it’s generally
close for shorter coils and most multi-layer coils, as I’ll
If we wind a coil on a magnetic material (such as a
ferrite rod), the inductance is greatly increased (Figure 3).
That’s because the permeability of the core is much
greater than the permeability of free space. The magnetic
core concentrates the magnetic field lines inside the coil
where they have the greatest effect.
Depending on the core material, the inductance
increase can be much more than a thousand, although
common radio frequency values range from about 200 for
VHF to 1,000 for lower frequencies.
The magnetic permeability of ferrite is large, so the
magnetic field lines are concentrated in the coil, thereby
increasing inductance by a large factor.
The inductance equation for such a coil is the same as
for an air-core coil multiplied by the relative permeability
K of the magnetic core:
L = N2Kµ0A/ℓ
The relative permeability is the ratio of the
permeability to the permeability of free space. The
magnetic core can be moved into and out of a coil to
make a variable inductor.
As an example that multi-layer coils closely follow this
same equation, I measured the inductance of the 52-turn
coil shown in Figure 3 and got 93 mH. When I put a
second layer of 42 turns over this, the inductance was 315
mH. Scaling by the ratio of the square of the turns gives
us a calculated inductance of 303 mH, which is just a few
Magnetic flux “leaks” out of each end of a short
solenoid coil, and that’s why the inductance equations
given above are specified for the ideal case of an infinite
A simple fix for such leakage is to join the ends
together to create a toroid, as shown in Figure 4.
The inductance of a single layer winding on a toroid is
L = N2Kµ0A/2pr
which is our familiar equation with the length replaced by
the unwrapped length of the toroid; that is, its
circumference at the midpoint, with the distance from the
center of the circle to the midpoint being the radius r.
Toroids offer a slightly greater inductance than short
solenoids for the same number of turns, since the
magnetic field lines are all contained inside the
While iron is a magnetic material, it doesn’t
make a good inductor core. That’s because it’s also
an electrical conductor, and the alternating currents
applied to a surrounding coil will lose energy by
heating the iron through induced eddy currents.
Think in terms of not putting metal objects into a
There’s also the problem that the iron will
“saturate;” that is, become so charged with a
magnetic field that it can’t be made to magnetize
There are various ways around this problem;
one of which is to load the iron with silicon to form
a steel alloy that’s less conducting, and to form the
core from thin sheets that are insulated from each
other. This is the method used to make cores for
most low frequency transformers. Another is to
make the iron into a fine powder and encapsulate
At radio frequencies, we need a better material
than iron or steel; one that won’t saturate and is an
46 July/August 2018
FIGURE 4. An inductor wound on a toroidal ferrite core.