Least Square Method-Based Unified General Design of Magnetic Field and Current Patterns for Improved Synthesized Magnetic Focusing (SMF)

A focused magnetic field is highly needed in medical applications such as transcranial magnetic stimulation (TMS) and implantable wireless power transfer (WPT). An optimum current pattern (CP) design of coaxial coils by appropriate selection of magnetic field pattern (MP) for synthesized magnetic focusing (SMF) is widely explored in this article. Using SMF equations, the CP of multiple coils (n) can be determined for any desired MP in theory. The magnitude of coil currents, however, could be extraordinarily large if inappropriate MP is sought. Furthermore, the MP for a few specific focal points (m) only could be controlled by the SMF theory; this means that the magnetic field for other points could not be controlled as wanted and could be arbitrary ( $m\le n$ ). The least square method is, therefore, first adopted in the SMF design for the case MP has a larger number of focal points than CP ( $n~\lt m$ ). Therefore, the number of focal points of MP m becomes arbitrary regardless of n due to the proposed unified general SMF design. The relationship between MP and CP for the innovative coaxial coils is specifically examined for the step, trapezoidal, and cosine-exponential MP cases. It is found that the smoother MP gives rise to a better CP with a minimum magnitude of currents and a minimum number of coils. Experiments for the cosine-exponential MP with three coils at the switching frequencies of $35\sim 75$ kHz show very good agreements with the simulation.