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.