Toroidal Helical Coils

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Toroidal helical coils have applications in electrical machinery such as transformers, motors, and antennas. This Demonstration shows some examples of the geometry of toroidal coils with helical windings.


The wire helices of the coil are called windings. Each loop of the wire is called a turn. The shape of the turns can be circular or square.

In this Demonstration, one winding of up to 48 turns or two windings of up to 24 turns are considered. The shape of the turn can be square or circular and the windings can be left- or right-handed.


Contributed by: Erik Mahieu (August 2015)
Open content licensed under CC BY-NC-SA



The parametric equations for the torus are


where is the radius of the torus ring and is the radius of the tube.

The parametric equation of a helix wound around the torus is


where is a constant defining the longitudinal offset of the helix from a zero reference, is the number of helical windings, and is the number of turns per winding.

The parameter is the winding direction: for right and for left.

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