Downhole neutron generators, downhole logging tools that utilize neutron generators, and methods to auto tune downhole neutron generators are disclosed. While a neutron generator is deployed in a borehole of a wellbore, the method includes determining whether an oscillation cycle of an ion beam current generated by the neutron generator is stable. After a determination that the oscillation cycle of the ion beam current is stable, the method includes determining proportional, integral, and derivative parameters of a proportional-integral-derivative controller that is operable to adjust an amount of power supplied to generate ions. The method further includes adjusting a replenish voltage of a replenish power supply of the neutron generator based on the proportional, integral, and derivative parameters.