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The Cygnus radiographic machine is a relatively compact low-energy (<3 MV) X-ray source with some extremely desirable features for radiographic applications. These features include small spot size, which is critical for high-spatial resolution, and high dose in a low-energy range. The X-ray source is based on bremsstrahlung production in a small-diameter (~0.75 mm) tungsten rod by a high-current (~60 kA) electron beam converging at the tip of the rod. For quantitative analysis of radiographic data, it is essential to determine the bremsstrahlung spectrum accurately. We have used the radiographic chain model to self-consistently model the diode with a 2-D particle-in-cell (PIC) code (Merlin) linked to an electron-photon Monte Carlo code to obtain the spectrum under three different situations: a steady-state spectrum using a voltage pulse of 2.25 MV, a time-integrated spectrum using a time-dependent experimental voltage pulse, and the spectrum resulting from inclusion of reflexing electrons around the anode rod in our PIC simulation. Detailed electron dynamics were obtained. We conclude that the time-integrated bremsstrahlung spectrum is significantly softer than that of the steady state. Including the effects of reflexing electrons using a Monte Carlo transport method in Merlin produced a spectrum in better agreement with experimental data.