Calculating the Square Coil

 

In the September 1, 2001 issue of the XSS Newsletter, Bill Simes 1 showed how to squeeze the most
inductance out of a length of wire when winding coils. Using Wheeler's formula, he calculated the
inductance for a given length of wire wound in a solenoid coil for different coil radii, finding that the
maximum inductance was obtained when the length of the coil was equal to the coil's radius. I then
remembered what Al Klase 2 had to say about coil efficiency, indicating that the sweet spot of coil
efficiency was when the coil length equaled its diameter. This led to me doing some calculating on my
own, thinking that a rearrangement of Wheeler's formula might more easily let you calculate an optimum coil
radius given the size of the wire, desired spacing between turns, and the desired inductance.

First, let's look at Wheeler's formula, and see if it can be made more "user friendly":

L = (n2 r2 )/(9r +10h) where L = inductance in microhenries (uH)
n = number of turns of wire
r = radius of coil in inches
h = length of coil

For a given inductance, there are still 3 variables, and you have to usually do several trial calculations
to arrive at the desired inductance. This can be a time consuming chore, even if you do like playing
with calculators or spreadsheets. Instead, let's try fixing the ratio of radius to length, and also settle
on the number of turns of wire per inch of length to see if we can make it a bit easier. For this, I
first let h = r to use Bill's maximum inductance scheme, and also did another set of figuring with h =2r to
match Al's recommendation. I also introduced another term T = turns per inch (tpi). T is pretty easy
to find or figure out. With enameled wire, tpi for various wire sizes are available, or you can figure it
out for yourself quickly if you have the wire already on hand by winding a close wound coil on any
convenient round form and then dividing the resulting number of turns by the length of the coil in inches.

Substituting into Wheeler's formula yields the following equations:

For maximum inductance (h = r)   r = (19L/T2) 1/3      

For a "square" coil (h = 2r)   r = ( 29L/4T2) 1/3    

Yeah, you have to take a cube root, but any scientific calculator makes it a snap. Now to see if any
insights are to be gained by doing all this rearranging. I selected coil values of 400 uH and 250 uH to
match up with the coils I usually end up making, the one covering the BC band with one of the little 200
pF polyfilm rf tuner capacitors (cheap and good for kid radios), and the other to go with the XSS 365 pF
variables. Shown below are my results, which agree closely with what I get with actual coils:

 

      T = 30 tpi (some wire donated for my students to use)                                                  

Method

 

L(uH)

 

r(in)

 

n (turns)

 

wire length (ft)

 

Max L

 

400

 

2.03

 

61

 

65

 

(h=r)

 

250

 

1.74

 

52

 

48

 

Square

 

400

 

1.48

 

89

 

68

 

(h = 2r)

 

250

 

1.26

 

75

 

50

 

These calculations show a larger diameter but shorter coil when maximizing inductance, with a savings
of a few feet of wire.

Adding Al's wisdom that for better coil efficiency use larger diameter wire and space winding, leaving
about 1 wire diameter between turns, I did a calculation for some #18 wire, using 10 tpi (with spacing)
for a 250 uH "square" coil, and found an optimum radius of 2.63 inches, 53 turns using 73 feet of
wire. Sounds pretty much like the coils used by the big scorers in some of the listening contests.

If your coil calculations give you a coil radius that is slightly different from that of the coil form you
had in mind, you can rearrange the equation to solve for T, using the radius of your form. The square
coil equation would then become:

T == (29L/4r3)1/2                                                      

This will get you very close to Al's optimum geometry. Multiply the resulting T by the coil length to
find the number of turns. Incidentally, when using the larger diameter wire, I typically use the plastic
coated stuff and let the coating thickness take care of the spacing. Optimum? Maybe not, but easy.

1 "Maximizing Coil Inductance", Bill Simes, XSS Newsletter Vol. 11, No. 5 of September 1, 2001

2   http://www.webex.net/~skywaves/Coils/Coilwiz.html