This paper designs a simple mathematical model for the voltage-controlled locally-active memristor (LAM), which undoubtedly simplifies the theoretical analysis. Owing to the N-shaped outline of the DC V-I curve, this LAM is called as N-shaped LAM. The small-signal analysis method is exploited to obtain the equivalent circuit of the N-shaped LAM when it is biased into the locally-active region. The calculation results show that the LAM can be equivalent to a conductance with a parasitic capacitor, which makes for a deep insight into the mechanisms at the origin of LAM based oscillator. The theoretical analysis reveals that the N-shaped LAM along with an inductor and a battery can generate oscillation, which is confirmed by the numerical simulation. In addition, observe from simulation results that as the inductance increases, the oscillation behavior of the system deviates from the sinusoidal signal and tends to a relaxation oscillation. Finally, the physical circuit realization of the N-shaped LAM based oscillator including the memristor emulator is proposed. The good agreement between the experimental results and simulation results further demonstrates the correctness and feasibility of the theoretical design and analysis.