Extension of Roth´s method to three-dimensional multi-region Eddy current problems

Roth´s method for solving two-dimensional boundary value problems has now been extended to 3- dimensional, two-reglon problems in which eddy currents can flow in one of the regions. This has been done by resolving the vector potential of the air-space into 2 parts: One part is called the limiting potential which exists when the eddy-current region is replaced by an infinitely permeable region. This potential is expressed as a triple infinite series in trigonometric functions, the current producing the field being represented by a triple Fourier coefficient. The second component is a modifying potential which together with the eddy currents in the second region has to satisfy all the boundary and interface conditions. All the field and current distributions are obtainedas complex triple infinite series. Tests carried on an experimental model show good agreement with the theoretical results. The theory indicates that the eddy currents in the region of finite ρ will be parallel to the interface between the two regions whatever may be the relative position of the exciting current. Possibilities of further applications of this method to 3-region problems, discontinuous boundaries and linear machines etc are also mentioned.