SYSTEMATIZATION OF THE FORMULAS OF THE RESONANT FERRITE ISOLATOR LOSS
Authors: Zaichenko O. B. , Zaichenko N. Ya. 
Published: April 2022

Context. The problem is to systematize and improve the models of a resonance ferrite isolator in the rectangular waveguide for the antenna-feeder devices, generating, receiving, measuring microwave equipment containing ferrite decoupling devices: ferrite isolators and circulators.
Objective. The goal of the work is to verify the formula for the losses of the resonant ferrite isolator in the direct and reverse directions, as well as the isolator ratio.
Method. The research method of the work is a critical analysis of literary sources, which was carried out, but did not bring the desired results, since it did not allow to verify the correctness of the derivation of the formula [17]. Therefore, a number of hypotheses were put forward, what the formula might mean. The difficulty lay in the presence in the formula of the product of trigonometric functions that can be attributed to frequency properties, which was taken as an initial hypothesis, which was not subsequently confirmed. The check included transformation of formulas using mathematical physics in terms of microwave electrodynamics, trigonometry and algebra. The beginning was the formula of the classics [16], similar to the formula of [18], accepted without proof. As it is known, for the main type of wave in a rectangular waveguide, the components of the magnetic field strength, obtained as a solution to the wave equation under the boundary conditions inherent in a rectangular waveguide. One component of the magnetic field strength is along the direction of wave propagation, and the second one is in the transverse direction in the section of the waveguide are proportional to the trigonometric functions cosine and sine with the same arguments. The equality of the two components of the strengths is traditionally uses to find the plane of circular polarization where to place the ferrite isolator, and so the authors use this proportionality to trigonometric functions in their derivation, namely the formulas of trigonometric functions of a double angle, the basic trigonometric identity sine squared plus cosine squared is equal to one for replacing the propagation constants with trigonometric functions, this allows to get rid of radicals in the formulas, these radicals in the formula are due to the phenomenon of dispersion in a rectangular waveguide. The rest of the manipulations with the formula are the reduction of similar terms.
Results. There was obtained analytical expressions for the losses of the resonant ferrite isolator in the forward and reverse directions, as well as the isolator ratio by strict mathematical transformations. There was performed such transformations. The ratios of the longitudinal propagation constant to the transverse propagation constant are replaced by the ratios of the trigonometric functions sine and cosine, since they are continuous as opposed to tangents and cotangents. Such a transformation allows to avoid square roots in the formula for the losses of the ferrite isolator in the forward and reverse directions, which are associated with the presence of dispersion in the waveguide, as in the formula for wavelength in the waveguide. The conversion is based on microwave electrodynamics. The formulas are used for the distribution of fields in a rectangular waveguide for the main type of wave. Further transformations consist in taking the common factor out of brackets and other arithmetic transformations.
Conclusions. Тhrer was obtained results partially coincide with the well-known [17], the derivation of the formula [17] was obtained for the first time, the studies carried out allowed us to reject the hypothesis that the product of cosines and sines in the loss formula of a ferrite isolator is a frequency characteristic, it appears as a result of arithmetic transformations. To take into account the frequency range, it is used that there is circular polarization at the middle frequency, there will also be circular polarization at the extreme frequency of the range, but the plane of circular polarization will shift in comparison with the position of the plane of circular polarization at the middle frequency. That is, a peculiar system of two equations is obtained with respect to two positions of the polarization plane relative to the wide side of the rectangular waveguide section. The scientific novelty consists in systematization and generalization of the formulas of the loss of the resonance ferrite isolator, the connection between the formulas from different literature sources, both foreign and domestic, is proved, which saves time for researchers of ferrite isolators for the verification of the formula. The practical significance. It may be useful for teaching purposes and in optimization of the ferrite isolator design.