Photoconductive antennae are one of the most common sources of terahertz radiation. Although extensive experimental and simulation work has been done to analyze the radiated fields, very limited theoretical analysis has been done to build elegant and compact models. This work presents the derivation of the radiated fields from a photoconductive antenna incorporating the effect of the semiconductor substrate. Additionally, an improved model of the semiconductor carrier dynamics has been proposed to analyze the radiation characteristics of the photoconductive antenna. The proposed carrier dynamics includes a novel analysis of the transient behavior of the substrate temperature and its effect on the parameters of the substrate material. It also incorporates the effect of the lifetime, polarization and velocities of the carriers as well as the effects of the radiated near zone fields and antenna geometry. The derived formulae and the proposed carrier dynamics model have been validated by comparing the results with the available experimental results in the literature. It has been found that the analytically calculated results closely follow the experimental results reported in the literature. © 2019 IOP Publishing Ltd.