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A mathematical formulation of the linear magnetoelectric effect was included in Lev Landau and Evgeny Lifshitz's Course of Theoretical Physics. [7] Only in 1959 did Igor Dzyaloshinskii, [8] using an elegant symmetry argument, derive the form of a linear magnetoelectric coupling in chromium(III) oxide (Cr 2 O 3).
In filters with cross couplings, it is convenient to characterize all filter couplings as a whole using a coupling matrix of dimension ,. [ 4 ] [ 12 ] It is symmetrical. Every its off-diagonal element M i j {\displaystyle M_{ij}} is the coupling coefficient of i th and j th resonators k i j . {\displaystyle k_{ij}.}
These equations are inhomogeneous versions of the wave equation, with the terms on the right side of the equation serving as the source functions for the wave. As with any wave equation, these equations lead to two types of solution: advanced potentials (which are related to the configuration of the sources at future points in time), and ...
To obtain the tensorial optical metric, medium properties such as permittivity, permeability, and magnetoelectric couplings must first be promoted to 4-dimensional covariant tensors, and the electrodynamics of light propagation through such media residing within a background space-time must also be expressed in a compatible 4-dimensional way.
In this way, such properties as the electric polarization, orbital magnetization, anomalous Hall conductivity, and orbital magnetoelectric coupling can be expressed in terms of Berry phases, connections, and curvatures. [5] [7] [8]
The coupling coefficient is the ratio of the open-circuit actual voltage ratio to the ratio that would be obtained if all the flux coupled from one magnetic circuit to the other. The coupling coefficient is related to mutual inductance and self inductances in the following way.
The coefficient of coupling k defines how closely the two circuits are coupled and is given by the equation = where M is the mutual inductance of the circuits and L p and L s are the inductances of the primary and secondary circuits, respectively.
Magnetic couplings can transmit power via a linear motion, rotary motion, or helical compound motion (a combination of linear motion and rotary motion). The combination of these transmission methods and different mechanical geometry can realize a wide variety of orderly motion in three-dimensional space.