Analysis of nonlinear gyromagnetic line operation using LLG equation

Abstract

This paper deals with the study of the gyromagnetic effect since it has proven to be an excellent way of generating radiofrequency (RF) using ferrite loaded nonlinear lines in the range of GHz. Using a proper mathematical model based on the analysis of the LLG (Landau-Lifshitz-Gilbert) equation will be demonstrated the dependence of the center frequency generated on both the intensity of the axial magnetic applied and the incident pulse amplitude.

Radio Frequency Oscillations in Gyrotropic Nonlinear Transmission Lines

“…In a number of experiments (e.g., [6,8,13,15]), the corresponding frequencies f 0 , f M , etc., lay below 1 GHz (with H 0 ranging from about 10 kA/m to 30 kA/m), while the recorded oscillation frequencies were between 0.3 GHz and 6 GHz. In fact, the magnitudes ω 0 , ω M etc, each having the dimension of frequency, appear in the form of combinations ω 0 + ω M , (ω 2 − ω 0 2 ), etc., in frequency-domain solutions for components of the magnetic permeability tensor (see [19]) that determine variations in the magnetization vector M. Note that in every specific experiment H 0 stayed constant, whereas H ϕ changed noticeably over the duration of the current pulse. A series of experiments and simulations [11,12,14] performed with 'triangular' pulse waveforms (see below, Figure 2) are, in this respect, of special interest.…”

“…The full set of wave modes supported by the transmission line is determined by its dispersion properties which are dictated by the line's geometry and size and the type of constituent relations for the filling material. The dynamics of variations in M can be described with the phenomenological Landau-Lifschitz (L-L) equation, e.g., taken in Gilbert's form [19]…”

“…Here γ = 2.8 × 10 10 Hz/T is the gyromagnetic ratio and H denotes the total magnetic field in the medium. The magnetic field H in Equation (1) is to be estimated from the Maxwell equations, with due account of all boundary conditions for the E and B fields, so as to 'automatically' involve the so called de-magnetizing terms owing to finite sizes of the ferrite beads and the core as a whole [19]. Next, α < 1 is a phenomenological parameter of this macroscopic equation, accounting for magnetic relaxation effects.…”

“…Accordingly, the formerly independent eigenmodes specified by the linear-approximation solutions for the E and B fields get coupled, too. A linear in h solution to Equation (3) yields the familiar magnetic permeability tensor µ [19]. With regard to pulse propagation, a major effect of nonlinearity is generation of higher-order harmonics by each component from the continuous spectrum presented by the pulse.…”

“…The excitation of RF oscillations in a coaxial ferrite-filled NLTL by a current pulse was analyzed by the present authors [10,12,14] with the help of a time-domain technique based on direct numerical integration of the Maxwell equation set, together with the Landau-Lifschitz equation of state for the ferromagnetic medium [19]. The computations were done in a version of the FDTD method [14], which for the time being is still limited by the assumption of n = 0 (a 2D + code involving z and ρ coordinates).…”