Further developments on Beth´s current-sheet theorem: computation of magnetic field, energy and mechanical stresses in the cross-section of particle accelerator magnets

The design of particle accelerator magnets involves computation of magnetic field and mechanical stresses distribution, together with a quench simulation. First approach is usually 2D, concerning the cross-section, while final calculations include the coil-end effect. Beth´s current theorem affords a very convenient way to obtain the magnetic field distribution, the magnetic field energy and the Lorentz force distribution, but with one important limitation: the coils must be replaced by a single current-sheet, which is a poor approximation if the coil is somewhat thick. Iron yoke is taken into account by the image method. However, saturation effect cannot be included. In this paper, a further development is proposed. Any winding can be considered as a sum of current sheets, placed at different radii. Results are obtained by the overall contribution: magnetic field distribution and harmonics at a reference radius, self-inductance, magnetic field energy and stress distribution. The main advantage of this analytical method is that the real coil geometry can be modeled. Besides, it is a fast and simple method for the first stage of magnet design, able to provide results not only about magnetic field distribution, but also about self-inductance and Lorentz forces. Calculations are made by means of a Matlab script, and are successfully compared with those obtained with other commercial packages based on FEM method or Biot-Savart´s law.