Piezoelectric energy harvesters are promising in the powering of wireless sensor networks with their ability to utilize ambiently available energy. A notable feature to be noted is the electromechanical coupling in the system. To explore the influence of tuning circuit, we develop here in this contribution a fully nonlinear model of a piezoelectric vibration energy harvester with a tuning circuit consisting of a resistor in parallel connection with an inductor. A linearized electromechanical model is obtained with small-amplitude base excitation. A series of numerical simulations are done to showcase the influence of the tuning circuit and uncover the parametric dependence of system output performances. It is concluded that the addition of connected inductor help improve the level of output voltage and power, while the working bandwidth of the system is widened to some degree. Advice is then provided upon the theoretical design and practical application of piezoelectric energy harvesters.