Exact expressions of the orbit-curvature and curvature-radius of the Toroidal/Helical orbits.

Quite recently, a number of multi-beam klystrons (MBK) have been developed experimentally, and at least three different MBK models are already commercially available. Multi-beam klystrons operate at reduced electron-gun voltage, and higher total beam-current than the single-beam designs, thus preventing occasional destructive gun- diode discharges, and increasing the power-efficiency up to ~ 75%. The power-efficiency (measured as the ratio of output microwave power to input DC power) is however still limited, even in MBK, as it is obviously impossible to extract all the microwave energy from a sharply-bunched electron-beam, without having the high space-charge-density of the slowing beam force it into uncontrollable defocusing.