The two familiar coil formulas sit at the ends of a range. NI = 35 000/((L/D)+2) applies when the coil bore area is less than twice the part area (high fill), and NI = 45 000/(L/D) when it is at least ten times the part area (low fill). Most real coil shots land somewhere between those two, and the standards close the gap with a straight-line interpolation on the area ratio Y = coil bore area / part cross-sectional area.

Physically the interpolation reflects how far the part sits from the windings. Close in, the flux is dense and few ampere-turns are needed. Far out, the flux has spread across the bore and more magnetomotive force is required to reach the same field in the part. At Y = 2 the formula returns exactly the high fill answer; at Y = 10 it returns exactly the low fill answer.

The length-to-diameter ratio still governs the demagnetising field from the part’s own end poles, so L/D enters through both endpoint formulas and remains restricted to the 2 to 15 band. For a round part in a round coil, Y is simply the square of the diameter ratio; for anything else, compare true cross-sectional areas and use the greatest cross-sectional diagonal as D.

Interpolated ampere-turns carry the same ±10% tolerance as the endpoint formulas and the same obligation to prove field adequacy on the part itself with a Hall-effect probe or an artificial flaw shim.

Worked example

Part length	250 mm
Part diameter	50 mm
Coil bore diameter	100 mm
Coil turns	5
L/D used	5
Fill area ratio Y	4
Ampere-turns at low fill limit	9000 A-turn
Ampere-turns at high fill limit	5000 A-turn
Required ampere-turns	6000 A-turn
Ampere-turns, -10%	5400 A-turn
Ampere-turns, +10%	6600 A-turn
Coil current	1200 A
Current, lower tolerance	1080 A
Current, upper tolerance	1320 A
Coil positions needed	1

A 250 mm x 50 mm bar in a 100 mm bore coil. L/D = 250/50 = 5. Y = (100/50)² = 4, inside the 2 to 10 interpolation band. Endpoints: NI(low fill) = 45 000/5 = 9 000; NI(high fill) = 35 000/(5+2) = 5 000. Interpolating: NI = [5 000 x (10-4) + 9 000 x (4-2)]/8 = [30 000 + 18 000]/8 = 6 000 ampere-turns. Over 5 turns that is 1 200 A, with a ±10% band of 1 080 to 1 320 A. The 250 mm part fits inside one coil position.