Inductance of Coil and Solenoid Calculator

L = mu_0 x mu_r x N^2 x A / l, where N is the number of turns, A is the cross-sectional area, l is the coil length, mu_0 is the permeability of free space, and mu_r is the core material's relative permeability (1 for air). This formula assumes an ideal, infinitely long solenoid.

Wheeler's approximation is a widely used empirical formula, L (in microhenries) = N^2 r^2 / (9r + 10l), where r and l (coil radius and length) are in inches, developed to accurately estimate inductance for practical single-layer air-core coils, including shorter and squatter coils where the ideal long-solenoid formula breaks down.

Use the long-solenoid formula only when the coil is genuinely long and thin (length much greater than radius), for typical hobbyist and RF coils, which are often shorter and wider, Wheeler's approximation gives a much more accurate real-world estimate.

Each turn contributes its own magnetic flux, and because all turns are magnetically linked, the total flux linkage scales with N (turns) times the flux per turn (which itself scales with N due to the combined field of all turns), giving inductance proportional to N squared overall.

Air-core coils have mu_r = 1 (technically air is very close to vacuum permeability). Ferrite cores commonly used in RF chokes and transformers have mu_r typically ranging from about 10 to several thousand depending on the ferrite material, dramatically boosting inductance for the same winding.

Wheeler's original 1928 formula was derived and published using inch units with a numerical constant (9 and 10) that already incorporates the permeability of free space and unit conversion factors, giving inductance directly in microhenries. Using different units without converting first will give an incorrect result.

Wheeler's approximation is accurate to roughly 1% for single-layer coils with length greater than about 0.8 times the radius, and remains a reasonable estimate down to about 0.4 times the radius, for much shorter or more complex multilayer windings, more detailed formulas or field simulation are needed for high precision.

Both formulas assume a constant, linear permeability. For air-core coils this holds essentially exactly regardless of current. For coils with a magnetic core, the effective permeability drops at high current as the core approaches magnetic saturation, reducing actual inductance below what these formulas predict at high drive levels.

A solenoid specifically refers to a coil that is long and thin relative to its diameter, producing a nearly uniform magnetic field inside it, while coil is a more general term for any wound conductor. The long-solenoid formula is named for and most accurate on true solenoid-shaped windings.

In long-solenoid mode, radius and length are in millimeters and relative permeability is dimensionless, with inductance reported in microhenries. In Wheeler mode, radius and length are in inches (as Wheeler's formula requires), with inductance also reported in microhenries.