Effective L/D ratio for coil magnetisation

Every coil ampere-turn formula is driven by the length-to-diameter ratio L/D, because that ratio governs how strongly the part's own end poles oppose the applied longitudinal field. A long slender bar has widely separated poles and a weak demagnetising field; a short fat one has strong opposition and needs far more ampere-turns.

For a hollow part the outside diameter overstates the problem. What matters magnetically is the amount of iron in the cross-section, not the outline. The standards therefore replace D with an effective diameter: the diameter of a solid bar having the same metal cross-sectional area. Generally D_eff = 2 sqrt((A_total - A_hollow)/pi), which for a plain tube reduces to D_eff = sqrt(OD² - ID²). A 100 mm tube bored to 60 mm behaves like an 80 mm solid bar, not a 100 mm one, so it needs fewer ampere-turns than its outside diameter suggests.

The empirical coil formulas are only valid between L/D = 2 and L/D = 15. Above 15 there is no further reduction in required ampere-turns, so 15 is used. Below 2 the demagnetising field swamps the applied field and the formula fails; the usual fix is to butt ferromagnetic pole pieces of similar diameter against the ends of the part to lengthen the magnetic circuit until the effective L/D reaches at least 2.

The same effective diameter is used for the fill factor decision, since fill factor compares the coil bore area with the part's true cross-sectional area.

D_eff = sqrt(OD² - ID²)      (plain hollow cylinder)
D_eff = 2 sqrt((A_total - A_hollow) / pi)      (general form)
L/D = part length / D_eff, used only within the range 2 to 15
Pole piece per end = max(0, (2 x D_eff - L) / 2)