Inductance Calculator - Solenoid and Toroid

Inductance L measures how strongly a coil opposes changes in current. A higher inductance means the coil stores more magnetic energy per amp of current and resists faster.

For a solenoid (a straight coil of N turns, length l, cross-section area A):

L = mu0 * mu_r * N^2 * A / l

mu0 = 4pi x 10^-7 H/m (permeability of free space). mu_r is the relative permeability of the core – 1 for air, around 200-10,000 for silicon steel or ferrite cores. The N^2 dependence is why doubling the number of turns quadruples the inductance.

For a toroid (a donut-shaped coil, mean radius R, cross-section area A):

L = mu0 * mu_r * N^2 * A / (2pi * R)

Toroids are preferred in power electronics because the closed magnetic path contains the flux, reducing EMI. Air-core toroids (mu_r = 1) are used in RF applications where a predictable, stable inductance matters more than high inductance per turn.

Inductive reactance at frequency f is XL = 2pi * f * L (in ohms). This is what makes inductors block high-frequency signals while passing DC – the foundation of low-pass filters and power supply chokes.

Energy stored in the magnetic field when carrying current I: U = 0.5 * L * I^2 (in joules). The chart shows inductive reactance vs frequency, showing how an inductor becomes increasingly resistive to higher frequencies.

One caveat that matters if you are winding a real coil. The solenoid formula above assumes an infinitely long coil, so the field is uniform inside and zero outside. A real coil leaks flux at both ends, and the shorter and fatter it is, the more it leaks. The formula therefore always reads high. For a coil roughly nine times longer than it is wide the overestimate is only about 5 percent, but for a squat coil the same length as its diameter it can be 30 percent or worse.

For single-layer air-core coils, which is most RF and antenna work, use the Wheeler formula instead: it is an empirical fit that includes the end correction and needs the coil diameter rather than its area. The toroid formula does not suffer from this at all, because the magnetic path closes on itself and there are no ends to leak from. That closed path is a large part of why toroids are chosen for power supplies.