Open-circuit　faults　in　power　switching　elements　are　primary　failures　of　active　magnetic　bearing　(AMB)　system.　However,　the　characteristics　of　current　signals　of　switching　elements　have　not　been　fully　used　in　the　diagnosis　of　open-circuit　faults　in　magnetic　bearings.　This　article　proposes　a　tracking　differentiator　(TD)　algorithm,　referred　as　Fast-TD,　to　extract　differential　information　from　the　current　signal,　which　is　applied　to　the　fault　diagnosis　of　open-circuit　failures　in　magnetic　bearings.　Especially,　the　rigorous　convergence　proof　of　the　Fast-TD　algorithm　is　presented.　First,　when　a　switching　element　of　the　three-phase　full-bridge　in　AMB　system　is　open-circuited,　theoretical　analysis　demonstrates　the　feasibility　of　diagnosing　the　open-circuit　faults　in　power　switching　elements　in　real　time　by　analyzing　the　differential　signals　from　the　current　sensor.　Then,　the　Fast-TD　algorithm　for　extracting　the　differential　information　of　current　sensor　is　proposed　and　its　convergence　is　proven　by　demonstrating　the　convergence　trajectories　of　the　states　within　different　regions.　Next,　the　current　differential　extraction　method　via　Fast-TD　is　applied　to　the　fault　diagnosis　and　fault-tolerant　control　(FTC)　of　a　magnetic　system.　The　case　studies,　which　include　comparison　simulations　with　two　other　differentiators,　verify　the　effectiveness　of　the　proposed　Fast-TD　algorithm.