Electrons that are confined in a Penning discharge geometry by electric and magnetic fields are observed to gain energy from a superimposed rf field. By shaping the electrodes in the form of hyperboloids of revolution the prebreakdown motion can be accurately described by the Mathieu equation. Resonances are predicted when the frequency of the electron in the electrostatic well is a multiple of half the rf frequency. In the pressure range investigated (air at 10−2to 10−5Torr), values as low as 40‐V dc and 3‐V rf were observed at the first resonance under conditions where the Penning mechanism alone required over 400‐V dc. Between resonances the Mathieu equation predicts runaway energy gain at considerably higher values of the rf voltage, but breakdown is observed at values much lower than the prediction. The explanation is found in energy gain through collisions with gas atoms. The collision resets the phase of the motion and allows the electron to continually gain energy from the rf field. Breakdown curves are calculated for the regions between resonances and are found to agree with experiment.