Abstract The present work focuses on ambient vibration energy harvesting. Specifically, this article deals with bistable piezoelectric energy harvesters (PEHs) which exhibits a wider bandwidth than linear oscillators. These complex systems require an energy extraction circuit (EEC) to rectify their voltage to supply power to autonomous sensors. This EEC needs to be optimized in order to increase the harvested power and even the bandwidth of PEHs. Because of the complex dynamics of bistable PEHs, 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 PEH coupled to a passive and an active synchronous EEC: respectively the standard energy harvesting (SEH) circuit and the parallel synchronized switch harvesting on inductor (P-SSHI) circuit. Experimental measurements conducted on a custom bistable PEH 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 PEH. 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.