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We report the fabrication, characterization, and modeling of photoconductive antennas using 40 nm thin-film flakes of black phosphorus (BP) as the photoconductor and hexagonal boron nitride (hBN) as a capping layer to prevent oxidation of BP. Dipole antennas were fabricated on oxidized high-resistivity Si substrates, and BP and hBN flakes were picked up and transferred onto the antenna inside a nitrogen glovebox. The transfer matrix technique was used to optimize the thickness of BP and hBN for maximum absorption. BP flakes were aligned with the armchair axis along the anode-cathode gap of the antenna, with crystal orientation measured using reflection anisotropy. Photocurrent imaging under illumination with 100 fs pulses at 780 and 1560 nm showed a bias-dependent maximum photocurrent localized to the antenna gap with a peak photoconductivity of 1 (2) S/cm in the linear regime of bias for excitation at 780 (1560) nm. Photocurrent saturation in bias (pump fluence) occurred at approximately 1 V (0.25 mJ/cm$^2$). Device performance was modeled numerically by solving Maxwell's equations and the drift-diffusion equation to obtain the photocurrent density in response to pulsed laser excitation, which was largely in qualitative agreement with the experimental observations. THz output computed from surface current density suggests that BP THz PCA performance is at least comparable to more traditional devices based on low-temperature-grown GaAs. These devices represent a step toward high-performance THz photoconductive antennas using BP.