134 W.K. Ao and P. Reynolds (a) (b) Fig. 17.2 Eddy current density using method of image for moving conductive plate in y and x direction. (a) Single magnetic pole projections moving conductive plate iny direction. (b) Single magnetic pole projections moving conductive plate inx direction The same method can be used to express Jiright in terms of J.P/, as given by J .ileft/ x i DJ .P/ x . 2B x; y/i D vyBz 2 tan 1 y a 2B x b tan 1 yCa 2B x b Ctan 1 yCa 2B x Cb tan 1 y a 2B xCb i (17.10) When the updated Eddy current density is obtained, it can be substituted into Eq. (17.4) to calculate the electromagnetic force byJ0x. Hence, the force can be written as Fem;y DFem;y j D Z Jx0 BzdVj D t Z a a Z b b Jx .P/ J x .iright/ J x .ileft/ Bzdxdyj (17.11) Similarly, moving the conductive plate in the x direction as shown in Fig. 17.2b, the method of image can also be used. Therefore, the total updated Eddy current density in the y component can be shown as follows J0y DJ0y j DJ .P/ y j J .iright/ y j J .ileft/ y j (17.12) where coordinate transformations can again be used to shift the primary Eddy current density J.P/ y to locations .x;2A y/ and .x; 2A y/, as given by Eqs. (17.13) and (17.14), respectively J .iright/ y j DJ .P/ y .x;2A y/j D vxBz 2 tan 1 x b 2A y a tan 1 x Cb 2A y a C tan 1 x b 2A yCa tan 1 x Cb 2A yCa j (17.13) J .ileft/ y j DJ .P/ y .x; 2A y/j D vxBz 2 tan 1 x b 2A y a tan 1 x Cb 2A y a Ctan 1 x b 2A yCa tan 1 x Cb 2A yCa j (17.14)
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