Optimising Current
Density
1,2,3
It
is possible to determine a wire diameter for a given coil geometry which will
maximise the current density in the coil. First we need to look at the circuit
as shown below 
Fig
1
R_{E}
is the external resistance (absolutely everything except the coil), R_{C} is the
coil resistance, VI is the induced voltage in the coil and Vs is the source voltage.
It is important to include source internal resistance and switching device resistance
when calculating RE. We'll keep things managable and concentrate on the steady
state condition so we can ignore Vs and VI as variables. First let's define the
current density J 
where
Ic is the current in the wire and Aw is the wire crosssectional area.
The
current density in the coil is then
where
F is the filling factor.
Applying
Kirchhoff's voltage law around the loop produces the following relationship 
Static
Current Analysis
Now
if we substitute the coil
resistance equation and eqn 3 above into eqn 2 we get 
This
is getting a bit messy so I'll lump together some of the constants 
giving
us 
This
is a fairly simple function and if we plot J against
we get the following general curve 
Fig
2
Obviously
only the +ve values of wire diameter have any meaning. This is quite revealing
since it shows that there is a certain value of
(for a given coil geometry) which will yield the maximum current density.
Let's
take the final step and tie down an expression which will give the wire diameter
for maximum current density without having to plot the whole function. If we differentiate
J we can get values for the turning points of the function. Using the Quotient
Rule gives us 

Eqn
6

The
differentiated function looks like this 
Fig
3
It
clearly has three roots and these can be found by equating J' = 0 
Now
apply some math manipulation 
The
expression we get above can now give us the roots. The numerator yields two expressions
for the roots
Solving
for these 
Only
the positive root is of interest. Now we can substitute for the constants C and
B as defined earlier to give 
The
wire diameter given by this expression is a good starting point from which to
begin experimentation. Keep in mind that the parameters which you use need to
be reasonably accurate for the calculation to give meaningful results. In particular
you'll need to produce an accurate value for the R_{E}. Whether you do
this by calculation or from measurement is up to you. One crucial aspect missing
from this analysis is the induced voltage term. Future analysis will concentrate
on developing an expression for optimised current density under dynamic conditions.
