# American Institute of Mathematical Sciences

• Previous Article
Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain
• NHM Home
• This Issue
• Next Article
Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks
June  2010, 5(2): 335-360. doi: 10.3934/nhm.2010.5.335

## Electromagnetic circuits

 1 Department of Mathematics, University of Utah, Salt Lake City UT 84112, United States 2 Institut de Mathématiques de Toulon, Université de Toulon et du Var, BP 132-83957 La Garde Cedex, France

Received  May 2009 Revised  February 2010 Published  May 2010

The electromagnetic analog of an elastic spring-mass network is constructed. These electromagnetic circuits offer the promise of manipulating electromagnetic fields in new ways, and linear electrical circuits correspond to a subclass of them. The electromagnetic circuits consist of thin triangular magnetic components joined at the edges by cylindrical dielectric components. Some of the edges can be terminal edges to which electric fields are applied. The response is measured in terms of the real or virtual free currents that are associated with the terminal edges. The relation between the terminal electric fields and the terminal free currents is governed by a symmetric complex matrix $\W$. In the case where all the terminal edges are disjoint, and the frequency is fixed, a complete characterization is obtained of all possible response matrices $\W$ both in the lossless and lossy cases. This is done by introducing a subclass of electromagnetic circuits, called electromagnetic ladder networks, which can realize the response matrix $\W$ of any other type of electromagnetic circuit with disjoint terminal edges. It is sketched how an electromagnetic ladder network, structured as a cubic network, can have a macroscopic electromagnetic continuum response which is non-Maxwellian, and novel.
Citation: Graeme W. Milton, Pierre Seppecher. Electromagnetic circuits. Networks and Heterogeneous Media, 2010, 5 (2) : 335-360. doi: 10.3934/nhm.2010.5.335
 [1] Hao Wang, Wei Yang, Yunqing Huang. An adaptive edge finite element method for the Maxwell's equations in metamaterials. Electronic Research Archive, 2020, 28 (2) : 961-976. doi: 10.3934/era.2020051 [2] Michel Lenczner. Homogenization of linear spatially periodic electronic circuits. Networks and Heterogeneous Media, 2006, 1 (3) : 467-494. doi: 10.3934/nhm.2006.1.467 [3] Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems and Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 [4] Ignacio García de la Vega, Ricardo Riaza. Bifurcation without parameters in circuits with memristors: A DAE approach. Conference Publications, 2015, 2015 (special) : 340-348. doi: 10.3934/proc.2015.0340 [5] Frank Jochmann. A singular limit in a nonlinear problem arising in electromagnetism. Communications on Pure and Applied Analysis, 2011, 10 (2) : 541-559. doi: 10.3934/cpaa.2011.10.541 [6] Denis Serre. Non-linear electromagnetism and special relativity. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435 [7] Fernando Miranda, José-Francisco Rodrigues, Lisa Santos. On a p-curl system arising in electromagnetism. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 605-629. doi: 10.3934/dcdss.2012.5.605 [8] Flaviano Battelli, Michal Fečkan. On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3043-3061. doi: 10.3934/dcdsb.2017162 [9] Juan Manuel Pastor, Javier García-Algarra, José M. Iriondo, José J. Ramasco, Javier Galeano. Dragging in mutualistic networks. Networks and Heterogeneous Media, 2015, 10 (1) : 37-52. doi: 10.3934/nhm.2015.10.37 [10] Radu C. Cascaval, Ciro D'Apice, Maria Pia D'Arienzo, Rosanna Manzo. Flow optimization in vascular networks. Mathematical Biosciences & Engineering, 2017, 14 (3) : 607-624. doi: 10.3934/mbe.2017035 [11] Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks and Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41 [12] A. Marigo. Robustness of square networks. Networks and Heterogeneous Media, 2009, 4 (3) : 537-575. doi: 10.3934/nhm.2009.4.537 [13] Manisha Pujari, Rushed Kanawati. Link prediction in multiplex networks. Networks and Heterogeneous Media, 2015, 10 (1) : 17-35. doi: 10.3934/nhm.2015.10.17 [14] Yi Ming Zou. Dynamics of boolean networks. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1629-1640. doi: 10.3934/dcdss.2011.4.1629 [15] Werner Creixell, Juan Carlos Losada, Tomás Arredondo, Patricio Olivares, Rosa María Benito. Serendipity in social networks. Networks and Heterogeneous Media, 2012, 7 (3) : 363-371. doi: 10.3934/nhm.2012.7.363 [16] H. N. Mhaskar, T. Poggio. Function approximation by deep networks. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4085-4095. doi: 10.3934/cpaa.2020181 [17] Jesse Collingwood, Robert D. Foley, David R. McDonald. Networks with cascading overloads. Journal of Industrial and Management Optimization, 2012, 8 (4) : 877-894. doi: 10.3934/jimo.2012.8.877 [18] Mauro Garavello. A review of conservation laws on networks. Networks and Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565 [19] Xiaoxian Tang, Jie Wang. Bistability of sequestration networks. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1337-1357. doi: 10.3934/dcdsb.2020165 [20] Yuki Kumagai. Social networks and global transactions. Journal of Dynamics and Games, 2019, 6 (3) : 211-219. doi: 10.3934/jdg.2019015

2021 Impact Factor: 1.41