Magnetic field due to helical currents on a torus

By means of coordinate transformations on different forms of the Biot-Savart law, general expressions are derived for both the magnetic vector potential and the magnetic field in two systems of toroidal coordinates. For the cases of interest, it is shown that the integrations can be performed exactly in proper toroidal coordinates, whereas approximations are necessary in quasi-toroidal coordinates. Different forms of helical windings are discussed. The results are applied to the calculation of a stellarator magnetic field of arbitrary polarity l, produced by a distribution of helical current filaments on a torus. The components of the magnetic field are given in terms of infinite series. In the intermediate region, away from the external separatrices and the central region, simple analytic expressions for the magnetic surfaces are obtained. (author)