An improvement in the calculation of the magnetic field for an arbitrary geometry coil with rectangular cross section

Hong Lei, Lian-Ze Wang and Zi-Niu Wu presented new simple and convenient solutions of the magnetic field for an arbitrary geometry coil with rectangular cross section. They treated two types of basic forms: the trapezoidal prism segment and curved prism segment. The curved prism segment has been divided into a series of small trapezoidal prism segments with the same cross section and its magnetic field is a vector sum of the individual fields created by each small trapezoidal prism conductor. For one trapezoidal prism conductor the magnetic field is obtained by 1-D integrals using Romberg numerical integration. In this paper, we give a completely analytical solution of these 1-D integrals that considerably saves the computational time especially in the computation of the magnetic field nearby the conductor surface, at the conductor surface and inside the conductor. From obtained analytical expressions the treatment of singularities can be easily done. Also, we tested four types of numerical integration (Gaussian, Romberg, Simpson and Lobatto) to find the most convenient singularity treatment of 1-D integrals. These obtained results are compared with those obtained analytically so that one can understand the advantage of the proposed approach. The paper points out on the accuracy and the computational cost.