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When a piezoelectric energy harvester is connected to a simple load circuit, the damping force which the piezoelectric element is able to generate is often below the optimal value to maximize electrical power generation. Circuits that aim to increase the power output of a piezoelectric energy harvester do so by modifying the voltage onto which the piezoelectric current source drives its charge. This paper presents a systematic analysis and comparison of all the principal types of power extraction circuit that allow this damping force to be increased, under both ideal and realistic constraints. Particular emphasis is placed on low-amplitude operation. A circuit called single-supply prebiasing is shown to harvest more power than previous approaches. Most of the analyzed circuits able to increase the power output do so by synchronously inverting or charging the piezoelectric capacitance through an inductor. For inductor Q factors greater than around only 2, the single-supply prebiasing circuit has the highest power density of the analyzed circuits. The absence of diodes in conduction paths, achievable with a minimum number of synchronous rectifiers, means that the input excitation amplitude is not required to overcome a minimum value before power can be extracted, making it particularly suitable for microscale applications or those with a wide variation in amplitude.