This paper proposes new sensorless vector control methods for permanent-magnet synchronous motors (PMSMs), which are based on a new minimum-order back-EMF state observer taking acceleration into account, and examines potentiality of the improved back-EMF observer through experiments. Conventional sensorless vector control methods for PMSMs using an estimate of back-EMF are established under the assumption of constant speed or zero acceleration, and consequently cannot operate properly at modes requiring rapid speed change especially in low speed region. On the other hand, the proposed back-EMF observer has the following features: (1) it is a new back-EMF state observer taking acceleration into account and requiring no additional approximation to motor mathematical model; (2) it is a minimum-order state observer; (3) it utilizes motor parameters in the simplest manner; (4) it can be applied to both salient-pole and non-salient-pole PMSMs; (5) it can be realized in both rotor and stator reference frames. Detailed designs and analyses for the improved state observer and the sensorless vector control systems in both rotor and stator reference frames are given. In relation to the sensorless vector control systems, this paper newly proposes a double-integral-type PLL method and an integral-feedback-type acceleration/speed estimation method. Their potentialities are examined through experiments.