This article presents a delay-suppressed sliding-mode observer (SMO) to observe the real-time rotor position of a permanent magnet synchronous machine (PMSM) controlled by vector control algorithms. First, in order to solve the low-pass filter (LPF) delay problem existing in the traditional signum function-based SMO, a brand new hyperbolic function is initially selected as the switching function. Because a hyperbolic function with a proper boundary layer is capable of reducing the chattering phenomenon of an SMO, it is not necessary to reemploy LPFs to eliminate the adverse impacts of chattering on the position estimation accuracy. In order to ensure the reachability and stability of the hyperbolic-function-based SMO, the observer gain is calculated by the means of a Lyapunov function in this article. Second, to solve the problem of calculation delay caused by digital computation, a current precompensation scheme based on dual-sampling strategy in one switching period is proposed. After compensating the calculation delay, the accuracy of position estimation as well as the motor control performance can be improved. Finally, the proposed SMOs with and without delay compensation are verified by both simulation and experiments that are conducted on a three-phase 1.5-kW PMSM drive prototype.