Large signal analytical study of bunching in klystrons

An analytical, nonlinear space charge wave theory is given for the velocity-modulated confined electron beam of finite radial extension. The present theory gives a more accurate nonlinear description of the electron velocities and the ac current amplitudes at the fundamental and harmonic frequencies than is given by earlier space charge wave-type theories. Although the theory is not strictly valid after electron overtaking has occurred, a formulation is used which allows the electron velocities to be multi-valued functions of the space coordinates. Therefore reasonably good results should also be expected in the overtaking range. For very large signals the results transform to those of Webster's ballistic theory, while for small signals the usual linearized space charge wave theory is obtained.