Solenoid Inductance Calculator
Most solenoid calculators give you one number from one fitted formula. This one evaluates the exact current-sheet integral and shows Wheeler 1928 next to it, so the spread between the two tells you how much the geometry is straining the closed form.
Choosing something to solve for makes Inductance (exact current sheet) an editable target and computes the chosen input from it.
Formula
| L | inductance (H) |
| N | number of turns |
| D, r | coil diameter and radius to the wire centres (m) |
| l | winding length (m) |
| kL | Nagaoka coefficient, 0 < kL <= 1 |
| mu0 | permeability of vacuum |
What this model assumes, and where it stops
Assumptions
- Single layer, air core, uniformly wound over the full length.
- The winding is idealised as a continuous cylindrical current sheet.
- Turns are circular and coaxial; no pitch angle, no lead-out geometry.
- Low frequency: no self-capacitance, no skin or proximity effect in the inductance itself.
- Nothing magnetic or conductive nearby.
Limitations
- The current-sheet idealisation ignores that real wire is discrete. For close-wound fine wire the error is well under 1%; for a few widely spaced fat turns it grows, and the Rosa round-wire corrections (not applied here) would be needed.
- Wheeler 1928 is quoted to about 1% only for coils longer than 0.4 diameters. Outside that band the deviation output will show it.
- Self-resonance is not modelled. Above roughly a tenth of the self-resonant frequency the measured inductance rises away from this value.
- A magnetic core changes everything here. This calculator is air-core only.
- A conductive former, chassis or shield can nearby reduce the inductance measurably; none of that is represented.
When you need a 3D field solution instead
Closed-form models like the one above hold on idealised geometry. These are the cases where they stop being good enough and a full 3D electromagnetic and thermal solution is the only way to get a trustworthy answer:
- Multi-layer or bank-wound coils, where layer-to-layer coupling and capacitance dominate.
- Any coil on a magnetic core, especially a gapped one - fringing flux around the gap is a 3D problem.
- Coils inside a shield can or near a ground plane, where image currents cut the inductance.
- Coupled coils and transformers, where it is the leakage inductance you actually need.
- Operation near self-resonance, or anywhere the AC resistance and inductance must be solved together.
- Non-circular, tapered, or partially wound formers that the current-sheet model cannot express.
Common questions
Which formula is more accurate, Wheeler or Nagaoka?
The Nagaoka current-sheet result is exact for the idealised geometry; Wheeler 1928 is a fitted approximation accurate to about 1% for coils longer than 0.4 diameters. This page shows both, and the deviation between them tells you how hard the approximation is working for your shape.
Should I measure the coil diameter over the wire or to the wire centres?
To the wire centres, which is the former diameter plus one wire diameter. The current-sheet model places the current at the wire centreline.
Why does my measured inductance differ from the calculated value?
The usual causes are self-capacitance if you measure near self-resonance, a conductive former or nearby chassis reducing the inductance, and the current-sheet idealisation of discrete turns. For close-wound fine wire the model error is well under 1%; for a few fat widely spaced turns it grows.
References
- Nagaoka - The Inductance Coefficients of Solenoids, J. Coll. Sci. Tokyo 27 (1909)
- Lorenz / Maxwell - Current-sheet self-inductance via coaxial filament mutual inductance
- Wheeler - Simple Inductance Formulas for Radio Coils, Proc. IRE 16 (1928)
- Grover - Inductance Calculations, Dover 1962