The present work focuses on ambient vibration energy harvesting. Specifically, this article deals with bistable piezoelectric energy harvesters which exhibits a wider bandwidth than linear oscillators. These complex systems require an energy extraction circuit to rectify their voltage to supply power to autonomous sensors. This energy extraction circuit needs to be optimized in order to increase the harvested power and even the bandwidth of piezoelectric energy harvesters. Because of the complex dynamics of bistable piezoelectric energy harvesters, there is a lack of simple and physically-insightful models in the literature that would allow the understanding and optimization of the extraction circuit. To address this issue, the present work derives closed-form models of a bistable piezoelectric energy harvester coupled to a passive and an active synchronous energy extraction circuit: respectively the standard energy harvesting circuit (SEH) and the parallel synchronized switch harvesting on inductor circuit (P-SSHI). Experimental measurements conducted on a custom bistable piezoelectric energy harvester demonstrate the validity of the proposed models with a relative error lower than 15% on the harvested power and the bandwidth. The proposed models allow to easily understand the influence of the P-SSHI circuit on the dynamics of a bistable piezoelectric energy harvester. Moreover, a comparison of the performance of the SEH and the P-SSHI circuits, valid for any bistable generator, is proposed. The latter shows that under low electromechanical coupling and low acceleration amplitude the P-SSHI circuit leads to multiply the maximum harvested power up to 4.3 compared to the SEH circuit, and the bandwidth by a factor of 2.3.