1344 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



111 this analysis, certain simplifications are made. One underlying 

 simplification is that of linearity; it is assumed that at low signal levels 

 the behavior of the electron stream, which is inherently non-linear, can 

 be represented by that of a truly linear system. 



As this paper purports to give an accurate and useful picture of the 

 low-level operation of microwave beam devices rather than an exhaustive 

 discussion, some details have been omitted or passed over lightly because 

 they seemed to be of secondary importance. Material which may be un- 

 familiar to workers in other fields but which is important as background 

 is presented in appendices A to C. Various points can be pursued further 

 in the literature. References to publications and to those responsible for 

 various advances are not given in the body of the paper or in the ap- 

 pendices; they are given for each topic in Appendix D. 



SPACE-CHARGE WAVES 



Many microwave tubes embody a long, narrow electron beam sur- 

 rounded by a conducting tube and focused or confined by a longitudinal 

 magnetic field. At low levels of operation, the radio-frequency disturb- 

 ances on such an electron stream can be expressed in terms of space- 

 charge waves. 



In these waves, two forms of energy are of primary importance: 

 electrostatic energy associated with the bunching together of electrons, 

 and kinetic energy, associated mth differences in the velocities of the 

 electrons. Thus, the waves may be called electromechanical; the electric 

 energy which we associate with waves in transmission lines and wave- 

 guides is present, but the magnetic energy is replaced by kinetic energy. 

 In circuit terms, we have an electrical capacitive element, but the in- 

 ductive element is inertial, not magnetic in nature. When the electron 

 and wave velocities are slow compared with the velocity of light, the 

 magnetic fields produced by the electron convection current are negli- 

 gible. 



There may be many space-charge modes or waves in an electron 

 stream, some with complex radial and angidar variations of amplitude 

 over the electron stream. Two waves predominate in the operation of 

 tubes, however, and one simplification we mil make is to deal mth these 

 only, and to disregard other modes of propagation on the electron stream. 

 Appendix A discusses such a pair of waves in a simplified physical system. 



We can associate with these two waves an ac electron convection cur- 

 rent i and an ac electron velocity v. Either we can assume that the elec- 

 tron beam is narrow and disregard the fact that these quantities vary 



