Re: How to construct an ideal Induction coil ?

Hi Birol,

I presume your application for the induction coil is to generate high
voltages from low voltage pulses.  In this case, your induction coil simply
consists of running a current through a primary coil with few windings,
cutting the current suddenly, and then using the large voltage generated
across the secondary winding for some purpose (a Jacob's ladder, for example).

The primary reason why induction coils are built on a bar shaped core
rather than a torus is simply an engineering proposition: It is easy to
wind a coil around a bar, and very difficult to thread it through a torus.
 This is because the secondary coil typically requires many kilometers of
wire, and to wind this around a torus would require threading a
multi-kilometer length of wire through the center of the torus millions of
times.  Compare this with mounting a rod on a lathe or similar device, and
allowing the wire to wrap itself around a spinning core in a few minutes. 
In addition, the insulation in the secondary is often non-trivial because
of the magnitude of the voltages involved.  Typically the voltage drop
between neighboring turns is very tiny, but the voltage drop between
neighboring layers of turns can be huge: if you have a 1/100 ohm drop in a
given turn, but 10000 turns/layer, then you have a 100 volt drop between
layers.  For reliability over time, you need to provide a good insulation
between the layers.

However, you are absolutely correct that a bar shaped core is very
inefficient.  The magnetic flux is poorly contained due to fringing at the
ends of the coil and the air does not store much energy and therefore has a
low inductance.  The good news is that you can make up for the loss by
using many more turns in your secondary, by increasing the lateral
dimensions of your bar so as to increase the total flux storage of the
device, and by increasing dI/dt in the primary.  Remember, the flux storage
is not only a function of the permeability of the material and the primary
coil's current, but also typically the cross-sectional area of the core.

If the air-path is philosophically troublesome for you, then you can make
four cores with ends cut at 45 degrees.  You can then join them into a
single square providing a closed loop for the magnetic flux.  You will
still loose some flux at the corners since it will be hard to get a perfect
fit, but you will nevertheless increase your efficiency dramatically.

However, a note about saturation.  As you know, the reluctance is inversely
proportional to the permeability of the material.  Therefore, to minimize
reluctance, you will be using a high permeability material -- hence your
choice of permalloy.  However, permalloy is a ferromagnetic material and it
will saturate.  If the inducing magnetic field (H) created by your primary
coil is sufficient to produce an induced field of > 1/3 Tesla or so, then
you will begin to see non-linear effects in the induced field.  You won't
be able to double the stored energy by doubling the inducing field anymore.
 By 2/3 of a Tesla, you won't be able to increase the magnetization of the
material no matter how much you increase your inducing field.  In this
case, your best option is to make a bigger core, or increase the number of
windings on your secondary.  Note, that if you have saturated the core, you
won't get much advantage from a torus shape if the cross-section of the
torus is identical to the cross-section of your core.  This is because the
total flux is limited at saturation.

In light of saturation, permalloy may not be your best choice of material.
 While it has superb permeability, it saturates at 2/3 of a tesla.  Compare
this with ordinary iron which saturates at approximately 1.5 Tesla.  Of
course, your inducing field must be stronger in order to create that field,
so you have to balance the ease and cost of increasing the number of turns
and the current through your primary coil, versus the cost and trouble of a
large core made from permalloy.

Finally, you wish to calculate the reluctance of the air path.  In this
case you're going to have to fall back on the definition of reluctance:

R = L/(A*mu)

where R is the reluctance, L is the length of the air path, A is the
cross-sectional area of the air path and mu is the relative permeability of
the air, which is 1.  You can create shells with infinitesimal thickness
and a cross-sectional area that encompasses all the flux, and the place the
surfaces perpendicular to the flux lines.  You can then integrate these
shells to obtain the total reluctance.  This is probably not a task for
doing by hand, but rather you should use a computer or find a way to
circumvent the problem (such as closing the magnetic loop or simply
building a beefy solenoidal core).

For an example of a simliar but simpler computation, see http://www.oz.net/~coilgun/theory/externaliron.htm.

You shouldn't be able to nullify the reluctance entirely since it is
inverse to the permeability.  You would need an infinite permability to
have a reluctance of zero.  That obviously won't happen, but indeed you are
right on the mark that the air path is by far the largest source of reluctance.

However, I would suggest that you simply design the primary to saturate the
core and build a core large enough to provide the flux you need to provide
the emf you wish to find in the secondary.  Also, you will want to consider
how to wind the coils because that is not easy!

A book references that I highly recommend is:

Griffiths, David J., Introduction to Electrodynamics, Third Edition,
Prentice Hall, New Jersey, 1999.

Griffiths is an excellent introduction to electrodynamics, and he has many
examples that will give you the theory you will need.

There are also several books written specifically on the manufacture of
induction coils in the late 1800's and early 1900's.  They may seem old,
but the technology is much the same with the exception that we now have
better materials.  For example:

Wright, Lewis, The induction coil in practical work including Rontgen X
rays, London, New York, Macmillan and co., limited, 1897.

I'd love to hear how you choose to design it and how it worked.  Let us know!

Zack Gainsforth