Fractional-slot winding configurations have attracted much attention due to the availability of concentrated windings and low cogging torque in permanent magnet brushless motors. For the design of the winding configurations, many design parameters must be determined. The winding factor provides a useful index for the optimal design. However, no general expressions of the winding factor have been derived for all the winding configurations. This paper performs the general formulation of the winding factor for the fractional-slot concentrated windings. The winding factor is redefined for stator windings without any information of the numbers of poles. For given stator windings, the optimal numbers of poles are determined from the obtained winding factors. The design strategy for the winding configurations is validated through a finite element method analysis. I peak Amplitude of phase current N Number of turns of a phase winding in a repeatable group N coil Number of turns in a coil Q Number of slots in a repeatable group β Ratio of coil pitch to the maximum pitch or π in winding degrees θ Angular coordinate in winding degrees θ 0 Phase of stator current φ coil Axis of a coil φ i Axis of the phase belt i φ i,j Axis of the coil (i, j) ν Harmonic order

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In this paper, we propose a control method for establishing periodic rotation inherent in parametric pendulum based on a delayed feedback control. The experiments elucidate the existing range of periodic rotation in the domain of delay. The range of existence possibly represents the tolerance of proposed control with mistuned delay. It is confirmed that forced synchronization governs the existence and the width. The result assures that the frequency synchronization characteristics overcome the mistuned difference of delay in the control through entrainment.

This paper analyzes frequency entrainment described by van der Pol and phase-locked loop (PLL) equations. The PLL equation represents the dynamics of a PLL circuit that appear in typical phase-locking phenomena. These two equations describe frequency entrainment by a periodic force. The entrainment originates from two different types of limit cycles: libration for the van der Pol equation and rotation for the PLL one. To explore the relationship between the geometry of limit cycles and the mechanism of entrainment, we investigate the entrainment using an energy balance relation. This relation is equivalent to the energy conservation law of dynamical systems with dissipation and input terms. We show response curves for the dc component, harmonic amplitude, phase difference, and energy supplied by a periodic force. The obtained curves indicate that the entrainments for the two equations have different features of supplied energy, and that the entrainment for the PLL equation possibly has the same mechanism as does the regulation of the phase difference for the van der Pol equation.

Parametrically excited pendulum inherently demonstrates a conversion from forcing vibration to rotational motion. The onset of periodic rotation strongly depends on the initial condition. We propose a control method to start up a parametric pendulum into periodic rotation based on an external force input with time delay. The feasibility of proposed method is verified numerically and experimentally. This paper advocates that the proposed method is suitable for crossing over a separatrix which governs the dynamics from initial conditions.