Efficient circuit representation of eddy-current fields

2017 ◽  
Vol 36 (5) ◽  
pp. 1457-1473 ◽  
Author(s):  
Yuji Shindo ◽  
Akihisa Kameari ◽  
Tetsuji Matsuo
Keyword(s):  
Analytical Solution ◽  
Orthogonal Polynomial ◽  
Eddy Current ◽  
Continued Fraction ◽  
Wide Frequency Range ◽  
Orthogonal Expansion ◽  
Current Field ◽  
Circuit Realization ◽  
Content Type ◽  
Circuit Representation

Purpose This paper aims to discuss the relationship between the continued fraction form of the analytical solution in the frequency domain, the orthogonal function expansion and their circuit realization to derive an efficient representation of the eddy-current field in the conducting sheet and wire/cylinder. Effective frequency ranges of representations are analytically derived. Design/methodology/approach The Cauer circuit representation is derived from the continued fraction form of analytical solution and from the orthogonal polynomial expansion. Simple circuit calculations give the upper frequency bounds where the truncated circuit and orthogonal expansion are applicable. Findings The Cauer circuit representation and the orthogonal polynomial expansions for the magnetic sheet in the E-mode and for the wire in the axial H-mode are derived. The upper frequency bound for the Cauer circuit is roughly proportional to N4 with N inductive elements, whereas the frequency bound for the finite element eddy-current analysis with uniform N elements is roughly proportional to N2. Practical implications The Cauer circuit representation is expected to provide an efficient homogenization method because it requires only several elements to describe the eddy-current field over a wide frequency range. Originality/value The applicable frequency ranges are analytically derived depending on the conductor geometry and on the truncation types.

Locating complex eigenvalues for analytical eddy-current models used to detect flaws

2019 ◽  
Vol 38 (6) ◽  
pp. 1800-1809
Author(s):  
Grzegorz Tytko ◽  
Łukasz Dawidowski
Keyword(s):  
Eddy Current ◽  
Design Methodology ◽  
Complex Function ◽  
Precise Determination ◽  
Argument Principle ◽  
Content Type ◽  
Conductive Material ◽  
Complex Roots ◽  
Complex Eigenvalues

Purpose Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function. Design/methodology/approach The proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them. Findings The elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time. Originality/value The paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

A nonlinear magnetic circuit model for periodic eddy current problems using T,ϕ-ϕ formulation

2017 ◽  
Vol 36 (3) ◽  
pp. 649-664 ◽  
Author(s):  
Rene Plasser ◽  
Gergely Koczka ◽  
Oszkár Bíró
Keyword(s):  
Eddy Current ◽  
Magnetic Circuit ◽  
Iteration Process ◽  
Equation System ◽  
Circuit Model ◽  
Computational Time ◽  
Combined Method ◽  
3D Fem ◽  
Content Type ◽  
Transformer Model

Purpose A transformer model is used as a benchmark for testing various methods to solve 3D nonlinear periodic eddy current problems. This paper aims to set up a nonlinear magnetic circuit problem to assess the solving procedure of the nonlinear equation system for determining the influence of various special techniques on the convergence of nonlinear iterations and hence the computational time. Design/methodology/approach Using the T,ϕ-ϕ formulation and the harmonic balance fixed-point approach, two techniques are investigated: the so-called “separate method” and the “combined method” for solving the equation system. When using the finite element method (FEM), the elapsed time for solving a problem is dominated by the conjugate gradient (CG) iteration process. The motivation for treating the equations of the voltage excitations separately from the rest of the equation system is to achieve a better-conditioned matrix system to determine the field quantities and hence a faster convergence of the CG process. Findings In fact, both methods are suitable for nonlinear computation, and for comparing the final results, the methods are equally good. Applying the combined method, the number of iterations to be executed to achieve a meaningful result is considerably less than using the separated method. Originality/value To facilitate a quick analysis, a simplified magnetic circuit model of the 3D problem was generated to assess how the different ways of solutions will affect the full 3D solving process. This investigation of a simple magnetic circuit problem to evaluate the benefits of computational methods provides the basis for considering this formulation in a 3D-FEM code for further investigation.

An analytical model for the behavior of I-shaped beam to cylindrical column connections at room temperature and high temperatures

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Seied Ahmad Hosseini ◽  
Mostafa Zeinoddini
Keyword(s):  
Analytical Solution ◽  
Elevated Temperatures ◽  
Element Model ◽  
Small Scale ◽  
Offshore Platforms ◽  
Rotational Stiffness ◽  
Rotation Curves ◽  
Content Type ◽  
Shaped Beam ◽  
Cylindrical Column

PurposeIn this paper, a closed-form analytical solution for the prediction of moment-rotation and the rotational stiffness-rotation curves of I-shaped beam to cylindrical column connections, commonly used on offshore platforms, at room and elevated temperatures, are presented.Design/methodology/approachAn analytical solution for the prediction of moment-rotation and the rotational stiffness-rotation curves of I-shaped beam to cylindrical column connections is presented. The results of this model are compared with those of a non-linear coupled mechanical-thermal finite element model and small-scale experimental tests previously provided by the authors.FindingsIn this paper, a closed-form analytical solution for the prediction of moment-rotation and the rotational stiffness-rotation curves of I-shaped beam to cylindrical column connections, commonly used on offshore platforms, at room and elevated temperatures, is presented. The required yield and plastic moments in this model are provided as an extension to Roark's relationships. The results of this model are compared with those of a non-linear coupled mechanical-thermal finite element model and small-scale experimental tests previously provided by the authors. A reasonable agreement has been found between the analytical model results and the experimental/numerical modeling results.Originality/valueThis article is extracted from the author’s doctoral thesis, and all its achievements belong to the authors of the article.

scholarly journals Analytical and Experimental Analyses of Nonlinear Vibrations in a Rotary Inverted Pendulum

2021 ◽  
Author(s):  
Mohammadjavad Rahimi dolatabad ◽  
Abdolreza Pasharavesh ◽  
Amir Ali Akbar Khayyat
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Closed Loop ◽  
Inverted Pendulum ◽  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Wide Frequency Range ◽  
Perturbative Method ◽  
Full State Feedback ◽  
Rotary Inverted Pendulum

Abstract Gaining insight into possible vibratory responses of dynamical systems around their stable equilibria is an essential step, which must be taken before their design and application. The results of such a study can significantly help prevent instability in closed-loop stabilized systems through avoiding the excitation of the system in the neighborhood of its resonance. This paper investigates nonlinear oscillations of a Rotary Inverted Pendulum (RIP) with a full-state feedback controller. Lagrange’s equations are employed to derive an accurate 2-DoF mathematical model, whose parameter values are extracted by both the measurement and 3D modeling of the real system components. Although the governing equations of a 2-DoF nonlinear system are difficult to solve, performing an analytical solution is of great importance, mostly because, compared to the numerical solution, the analytical solution can function as an accurate pattern. Additionally, the analytical solution is generally more appealing to engineers because their computational costs are less than those of the numerical solution. In this study, the perturbative method of multiple scales is used to obtain an analytical solution to the coupled nonlinear motion equations of the closed-loop system. Moreover, the parameters of the controller are determined, using the results of this solution. The findings reveal the existence of hardening- and softening-type resonances at the first and second vibrational modes, respectively. This led to a wide frequency range with moderately large-amplitude vibrations, which must be avoided when adjusting a time-varying set-point for the system. The analytical results of the nonlinear vibration of the RIP are verified by experimental measurements, and a very good agreement is observed between the results of both approaches.

Sign in / Sign up

Export Citation Format

Share Document