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Mathematical models of excitation of electrodynamic systems by an intensive electron beam


B.K. Sivyakov – Dr. Sc. (Eng.), Professor, Department «Electrotechnique and Electronics», Yuri Gagarin State Technical University of Saratov
E-mail: sibokon@
D.B. Sivyakov – Ph. D. (Eng.), Associate Professor, Department «Electrotechnique and Electronics», Yuri Gagarin State Technical University of Saratov
E-mail: sibokon@

Excitation by an intense electron beam of the electrodynamic system is widely used in modern and promising high-power microwave devices (traveling and backward wave tubes, klystrons, etc.), so the construction of mathematical models different from the widely known ones (Weinstein-Solntsev and Row J.E.).
A different approach to constructing mathematical models is proposed. In the electrodynamic theory of excitation, based on the re-presentation of the total electromagnetic field in a system with an electron beam in the form of eigenfunction series of an electrodynamic system without an electron beam (decomposition method). In this case, poor convergence of the obtained series is observed because of the features of the electric and magnetic total fields of the system with the electron flux at the boundaries of the electron beam (the jumps of the divergence of electric and the rotor of magnetic fields) and, as a consequence, the need for subsequent separation of the electronic fluxes from the resulting solution for their calculation by others methods. The problem is aggravated with an increase of the intensity of the electron beam in high-power electronic devices and, accordingly, of its fields.
To eliminate this problem, it is proposed to initially extract the external eigenfields of the electron beam from the total field and only then perform the expansion of the total field in a series of eigenvectors of the electrodynamic system.
Initially, the sources of the total field in the space of interaction with the electron beam are charge and current density of the electron beam, which excite the total electromagnetic field in the waveguide, described by the inhomogeneous system of Maxwell's equations.
The presence of charges in the flow is the source of its own potential electric field, which after the introduction of the potential is de-scribed by the Poisson equation. The non-uniform wave equation corresponds to the remaining after the extraction of the electric po-tential field of the solenoidal part of the electric field.
The convection current and displacement current of the intrinsic potential field are sources of the intrinsic magnetic field of the electron beam, which, after the introduction of the vector potential, obeys the generalized (vector) Poisson equation.
It is possible to continue to extract the proper fields of the electron beam. Thus, the source of its own solenoidal electric field is its own magnetic field, etc. However, these will be weaker fields. They are always described by Poisson's equations and, consequently, have quasistatic properties and can not independently exist and propagate independently of their sources in the electron beam.
In contrast to them, the parts of the total field that remain after separating the quasistatic fields contain electromagnetic fields that can independently exist and propagate independently of their sources in the electron beam. They are also described by the corresponding inhomogeneous wave equations. It is they who characterize the signal amplified or generated in the electronic device.
The separation of at least two of the main strongest electric and magnetic proper fields of the electron beam ensures the continuity of the first derivatives of the fields that expand later in the waveguide, which, according to the expansion theorem, is a condition for absolute and uniform convergence of series in eigenfunctions.
Subsequent application of the method of variation of arbitrary constants and the decomposition method made it possible to obtain a mathematical model of the excited field of the signal in an electronic device.
Taking into account the displacement current, it was possible to identify two effects that are not described by the well-known elec-trodynamic theory-the effect of the introduced coupling parameter on the excitation of an electromagnetic wave by current and the nonlocal nature of the coupling between the field and current, expressed in the effect of influence of the current derivatives on the excitation process. After the corresponding transformations of the waveguide excitation equation, the excitation equation for a one-dimensional nonlinear theory of a traveling-wave tube is obtained. Similar mathematical models are formulated for excitation of reso-nator systems by an intense electron beam in high-power microwave devices of superhigh frequencies.

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June 24, 2020
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