Coils and inductors

Air-core coil — a single-layer solenoid. The inductance is the current-sheet value L = µ₀·π·r²·N²·K/l with Nagaoka’s coefficient K from Lundin’s approximation (accurate to a few parts per million), minus Rosa’s correction for round wire: ΔL = µ₀·r·N·(Ks + Km), with Ks = 5/4 − ln(2p/d) for the wire’s own self-inductance and Km Rosa’s series for the mutual terms. Close-wound coils come out a little below the current sheet; widely spaced ones a little above. Wheeler’s formula, L(µH) = r²N²/(9r + 10l) in inches, is shown for comparison (within 1% when l > 0.8r). Turns for an inductance solves the same model for N, with the coil getting longer as turns are added. The diameter is measured to the wire centres (former + one wire diameter).

Self-resonance and Q — self-capacitance from Medhurst’s measurements as fitted by Knight, C(pF) = 0.1126·l + 0.08·D + 0.27·√(D³/l) in cm, a ±20% figure that ignores leads and surroundings. The AC resistance uses the two asymptotes of the round-wire skin-effect solution; proximity effect between turns usually adds another ×1.5–3, so the Q shown is an upper bound. Keep the working frequency below about a third of the self-resonant frequency.

Multilayer — Wheeler’s multilayer formula L(µH) = 0.8·a²N²/(6a + 9b + 10c) (a mean radius, b length, c depth, inches), good to about 1% for reasonably proportioned coils. The most inductance per length of wire comes from a Brooks coil: a square winding section with a mean diameter three times its depth.

Toroids & cores — L = AL·N². The core list has Micrometals iron-powder toroids (mixes 2, 6, 10, and 26; Micrometals quotes AL in µH per 100 turns) and Fair-Rite ferrite toroids sold as Amidon "FT" cores (mixes 43, 61, 77; AL in nH/N²), with catalogue AL values and dimensions. Enter AL by hand for anything else — pot cores, RM and E cores, beads. Peak flux density is B = L·I/(N·Ae); for ferrites it is compared with the saturation flux density of the material at 25 °C (lower when hot). Iron powder saturates gradually, so check the maker’s DC-bias and core-loss curves for power use. AL tolerances are about ±5% for iron powder and ±20–25% for ferrite: measure the wound part when it matters.

Magnet wire — AWG diameters (0.005 in × 92^36 − n)/39, an approximate single-build enamelled diameter, resistance at 20 °C (copper, 1.72 × 10⁻⁸ Ω·m), and the skin-effect AC/DC ratio at a frequency you choose. Skin depth in copper is about 66 µm at 1 MHz.

Energy & V = L·dI/dt — stored energy ½LI², the current change from a voltage applied for a time, and the voltage needed to change a current quickly: the kick a relay or motor coil gives without a flyback diode.