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June  2010, 5(2): 335-360. doi: 10.3934/nhm.2010.5.335

Electromagnetic circuits

Graeme W. Milton 1, and  Pierre Seppecher 2,

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.
Keywords: Electromagnetism, Circuits, Metamaterials., Networks.
Mathematics Subject Classification: Primary: 35Q60, 78A48, 94C0.
Citation: Graeme W. Milton, Pierre Seppecher. Electromagnetic circuits. Networks & Heterogeneous Media, 2010, 5 (2) : 335-360. doi: 10.3934/nhm.2010.5.335
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