Introducing the virtual air-gap scheme to the Kelvin force densities with external and total field

Three different Kelvin force expressions were expanded by incorporating the virtual air-gap scheme to calculate the total force and the distributions of force density. Conventionally, the Kelvin force density can be expressed by using two different fields such as an external applied field and total field acting on a dipole. Until now, the Kelvin force density with total field (KV) has been used for calculating the total force and force density. Since there is no single well-accepted manipulating technique, it has been difficult to directly compute the Kelvin force by adopting the external field approach; however, using the virtual air-gap scheme, the direct calculation of both Kelvin force density with external applied field (KVE), and with total field (KVT) as the local distributions of body force density was able to be evaluated. We tested three numerical analysis models to verify our proposed method. First, the nonlinear magnetic material was considered and compared the total force with different gap distances to contact position. Second, the distributions of force density were compared to each other. An irregularity aroused in the distributions of force density by the KVE and KVT. Finally, to remove the irregularity, one possible approach, forming sliced-tube regions aligned with the field direction, was suggested and examined.