An inductor is a coil of wire. It stores energy in a magnetic field and resists changes in current, so it passes DC easily but opposes fast-changing signals. Wound on a magnetic core, the same wire gives far more inductance; wound for a different purpose, the core becomes a ferrite that soaks up unwanted RF. The theory is in Inductance.
Cores and winding
Without a core (an air-core coil), inductance depends on turns, diameter and length. A magnetic core multiplies inductance, so fewer turns do the job. Two core materials do most of the work in radio:
Core
Typical use
Powdered iron
tuned circuits and filters: stable with temperature, good Q
Ferrite
transformers, chokes and EMI suppression: very high permeability
Different ferrite mixes work best in different frequency ranges, so the mix, not the size, must match your band.
Toroid: big inductance from few turns, field kept in the core, and the mix tuned to a frequency range.
A toroid (ring) keeps nearly all the magnetic field inside the core. Coils on a ring therefore couple little to their neighbours, which is why they suit compact RF circuits.
Toroid: field stays in the core. Ferrite bead: lossy at VHF/UHF, so it damps parasitics.
The core maker publishes an AL value (inductance per turn squared) so you can calculate the winding.
Count the wire passing through the hole, not the loops on the outside. Double the turns and the inductance quadruples.
Inductance5 µH
Ferrites as chokes
A ferrite bead or clamp-on core adds impedance in the path of RF current. It does nothing to the wanted signal travelling inside a cable, because the currents inside are equal and opposite and their fields cancel. It does oppose current flowing on the outside of the cable (common-mode). Cores can also work as suppressors of parasitic oscillation on amplifier leads.
The ferrite adds impedance to common-mode current only. The wanted signal passes untouched.
For how to wind or choose one for the antenna feed line, see Common-mode chokes.
Pitfalls
A real coil has a little capacitance between turns. Above its self-resonant frequency it behaves as a capacitor, not an inductor.
A core can saturate: with too much current the inductance collapses and the core heats up. Power chokes need cores sized for the current.
A ferrite in a transmitting path dissipates energy as heat. A core that is too small, or the wrong mix, can get hot at high power.