The performance of the converging electron beam generated in cylindrical triodes is systematically studied by particle-in-cell code simulations. Depending on the cathode and grid potentials applied, different operation regimes are identified. For low voltages between cathode and grid, laminar flow and homogeneous beam energy density at the target (anode) is obtained. This applies both to the case of unipolar electron flow and to bipolar flow with counter-streaming ions. Hereby, the electron emission current is enhanced by about 50% for bipolar flow compared with unipolar flow. A further increase by about 20% is obtained when electron backscattering at the target is enhanced due to a change of target material from aluminum to tungsten. For cathode-grid voltages exceeding a critical value, laminar flow is replaced by non-laminar flow regimes. For unipolar electron beams, a virtual cathode forms between grid and target, which leads to an inhomogeneous power density at the target. For the specific geometry investigated and the cathode potential fixed at −120 kV, the cathode-grid voltage threshold for the formation of the virtual cathode is ~32 kV for Al targets and ~28 kV for W targets. For bipolar flow, the laminar flow regime already ends at cathode-grid voltages of ~23 kV (Al target) and ~20 kV (W target), respectively, and is replaced by magnetic insulation at the beam edge. For increasing cathode-grid voltage, the magnetically insulated region extends until beam pinching occurs.