In this paper a comprehensive analytical analysis is performed based on a new accurate electrical model of silicon photomultiplier (SiPM) detectors. The proposed circuit model allows to accurately reproduce the SiPM output time response regardless of the particular technology adopted for the fabrication process, and can also be profitably exploited to perform reliable circuit-level simulations. A novel expression of the detector photoelectron response due to a single photon absorption is systematically developed. The obtained waveform accurately reproduces the fast detector ignition, the ensuing avalanche self-quenching and the final slow recharging operation. Predictive capabilities of the adopted analytical model are demonstrated by means of experimental measurements on a real SiPM device.