Coil Inductance
Single-layer air-core solenoid inductance from turns and geometry (Wheeler), or the turns for a target L.
μr = 1 for air or plastic cores. Wheeler is accurate to ~1% when l ≥ 0.4·d.
Wheeler's current-sheet formula
For a single-layer solenoid whose turns approximate a uniform current sheet, the inductance is
where is the number of turns, the coil diameter and the winding length (both in metres). The term is the end-effect correction that turns the ideal infinite-solenoid result into a finite-coil one. To hit a target inductance, solve for the turns:
The approximation stays within roughly 1% as long as the length is at least about 0.4 times the diameter. Use the relative permeability to account for a magnetic core — air is 1, while ferrite or iron-powder cores raise in direct proportion.
Frequently asked questions
How do I calculate the inductance of a coil?
For a single-layer air-core solenoid use Wheeler's current-sheet formula L = μ₀·μr·N²·A / (l + 0.45·d), with A = π·d²/4 the cross-section, d the coil diameter and l the winding length (in metres). A 10-turn coil 10 mm in diameter and 10 mm long gives A = 7.854e-5 m², l + 0.45d = 0.0145 m, so L ≈ 0.68 µH.
What is the Wheeler formula for inductance?
Wheeler's current-sheet approximation is L = μ₀·μr·N²·(π·d²/4) / (l + 0.45·d). The 0.45·d term corrects the ideal infinite-solenoid result for a finite coil; it stays within about 1% as long as the length l is at least 0.4 times the diameter d.
How many turns do I need for a target inductance?
Rearrange Wheeler's formula: N = √( L·(l + 0.45·d) / (μ₀·μr·A) ). For L = 1 µH on a coil of d = 10 mm and l = 10 mm in air, N ≈ √(1e-6·0.0145 / (1.2566e-6·7.854e-5)) ≈ 12.1, so you wind 13 turns and trim.
Does the core material change the inductance?
Yes — inductance scales directly with the relative permeability μr. Air or plastic has μr ≈ 1, while a ferrite or iron-powder core can multiply L by tens to thousands. Set μr to your core's effective permeability; note that gapped or saturating cores have an effective μr below the raw material figure.