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In this paper, we present several nonlinear control designs for a single one-degree-of-freedom (1-DOF) active magnetic bearing (AMB). The primary control objective is to globally asymptotically stabilize the mechanical states of an AMB while reducing the AMB power losses. This suggests operation with zero- or low-bias (ZB/LB) flux. We derive a flux-based model for an AMB using a generalized complementary flux condition. This condition is imposed both for ZB and LB operations. A convenient model structure results, in which the ZB mode is a special case of the more general LB mode of operation. We next derive control laws for LB and ZB AMB operation. The control designs borrow ideas from the theory of control Lyapunov functions (clfs) and passivity. The performance of each proposed control design is evaluated via numerical simulations with a high-fidelity AMB model. Implementation issues are also discussed.