Direct steady-state solutions for circuit models of nonlinear electromagnetic devices

Abstract

Purpose This paper aims to omit the difficulties of directly finding the periodic steady-state solutions for electromagnetic devices described by circuit models. Design/methodology/approach Determine the discrete integral operator of periodic functions and develop an iterative algorithm determining steady-state solutions by a multiplication of matrices only. Findings An alternative method to creating finite-difference relations directly determining steady-state solutions in the time domain. Research limitations/implications Reduction of software and hardware requirements for determining steady-states of electromagnetic. Practical implications A unified approach for directly finding steady-state solutions for ordinary nonlinear differential equations presented in the normal form. Originality/value Eliminate the necessity of solving high-order finite-difference equations for steady-state analysis of electromagnetic devices described by circuit models.

Archives of Electrical Engineering

“…To obtain the discrete differential operators relating the values of that function and its first partial derivatives, the relations between the values of the periodic function and its Fourier coefficients should be determined for the series in (17), when limited to a finite number of terms −𝑅 < 𝑟 < 𝑅 and −𝑆 < 𝑠 < 𝑆. For such a function, unique relations can be found between values of the function 𝑧(𝑥, 𝑦) and its Fourier coefficients 𝑍 𝑟 ,𝑠 , when selecting an arbitrary set of (2𝑅 + 1) • (2𝑆 + 1) points {𝑥 𝑛 , 𝑦 𝑛 }, where: 0 < 𝑥 𝑛 < 2𝜋 for 𝑛 ∈ {1, 2, .…”

“…The vectors Z 𝑥 and Z 𝑦 contain the Fourier coefficients of the respective partial derivatives from (17), and are ordered analogously as the Z vector.…”

“…The operators D (1) 𝑥 and D (1) 𝑦 can be developed based on the expressions defining a set of the Fourier coefficients of the series in (17).…”

“…Second-order discrete operators for the function 𝑧(𝑥, 𝑦) in (17), if the second-order partial derivatives exist, can be found based on the discrete operators developed in the previous chapters. Three discrete operators could be determined to three second-order partial derivatives when represented by their Fourier series:…”

“…The FDOs of periodic and two-periodic time functions are presented in [14] and [15] and have been successfully tested in [16,17] for steady-state analysis of electromagnetic circuits. The same methodology has been used in [18] to develop FDOs to solve boundary-value problems for Ordinary Differential Equations (ODEs).…”