On the Build-Up of Oscillations in a Cylindrical Magnetron

Abstract

A fully nonlinear theory of the magnetron is presented starting from the equations of classical electrodynamics. A crucial step in the development is an orthogonal-series expansion of the space charge and current densities using appropriate sets of cavity modes, the expansion coefficients being nonlinear functionals of the electron trajectories. Splitting the electric field vector into its solenoidal and irrotational (gradient of a scalar potential) parts, expanding the magnetic field and the solenoidal part of the electric field in a complete set of orthonormal solenoidal cavity modes, and substituting these expansions into Maxwell's equations, differential equations governing the time evolution of the mode amplitudes are derived. These differential equations for the mode amplitudes and the Poisson's equation for the scalar potential are solved in terms of the charge and current densities in the interaction region. Substitution of the explicit expression for the field components in terms of the charge and current densities into the equations of electron motion reduces the latter to a fixed-point problem for a vector nonlinear operator in an appropriate function space. The fixed point, and hence the solution for the oscillatory fields inside the magnetron, may be found by standard successive approximation techniques. For the special case of a smooth-anode magnetron (magnetron diode) the mode-amplitude equations are completely solved to yield the initial growth of the mode amplitudes. It is seen from this solution that as soon as the electrons are made to follow a curved path leading to a nonzero azimuthal current in the magnetron cavity, oscillations start to build up from a zero initial value. Thus it is seen, contrary to the conclusions of the current theories of the magnetron, that it is totally unnecessary to assume, a priori, the presence of any kind (random or deterministic) of high-frequency disturbances initially in the cavity to explain the build-up phenomenon. The detailed study of oscillation build-up based on the successive approximation solution for the electron trajectories will be deferred to the second part of this contribution.