  We explore the properties of run-and-tumble particles moving in a
piecewise-linear "ratchet" potential by deriving analytic results for the
system's steady-state probability density, current, entropy production rate,
extractable power, and thermodynamic efficiency. The ratchet's broken spatial
symmetry rectifies the particles' self-propelled motion, resulting in a
positive current that peaks at finite values of the diffusion strength, ratchet
height, and particle self-propulsion speed. Similar nonmonotonic behaviour is
also observed for the extractable power and efficiency. We find the optimal
apex position for generating maximum current varies with diffusion, and that
entropy production can have nonmonotonic dependence on diffusion. In
particular, for vanishing diffusion, entropy production remains finite when
particle self-propulsion is weaker than the ratchet force. Furthermore, power
extraction with near-perfect efficiency is achievable in certain parameter
regimes due to the simplifications afforded by modelling "dry" active
particles. In the final part, we derive mean first-passage times and splitting
probabilities for different boundary and initial conditions. This work connects
the study of work extraction from active matter with exactly solvable active
particle models and will therefore facilitate the design of active engines
through these analytic results.
