The positivity and other properties of the matrix of capacitance: Physical and mathematical implications

Díaz, Rodolfo A.; Herrera, William J.

doi:10.1016/j.elstat.2011.08.001

Cited by 15 publications

(14 citation statements)

References 26 publications

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“…In particular, we find that the leading-order properties of the resonant frequencies and associated eigenmodes are given in terms of the eigenstates of the generalized capacitance matrix, as introduced in [4]. This is a generalization of the notion of capacitance that is widely used in electrostatics to model the distributions of potential and charge in a system of conductors [16].…”

Section: The Generalized Capacitance Matrixmentioning

confidence: 91%

Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors

Davies¹,

Herren²

2021

Preprint

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The aim of this work is to derive precise formulas which describe how the properties of subwavelength devices are changed by the introduction of errors and imperfections. As a demonstrative example, we study a class of cochlea-inspired rainbow sensors. These are devices based on a graded array of subwavelength resonators which have been designed to mimic the frequency separation performed by the cochlea. We show that the device's properties (including its role as a signal filtering device) are stable with respect to small imperfections in the positions and sizes of the resonators. Additionally, if the number of resonators is sufficiently large, then the device's properties are stable under the removal of a resonator.

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Section: The Generalized Capacitance Matrixmentioning

confidence: 91%

Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors

Davies¹,

Herren²

2021

Preprint

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“…Here we have made the redefinition γ i c ii → γ since c ii is always positive as well 24 25 . This update does not require the sum over N terms, and does not explicitly require knowledge of c ii .…”

Section: Methodsmentioning

confidence: 99%

“…Since c ij is a positive matrix 24 25 , the quantity is always positive. Therefore, the total error is always decreasing provided γ is chosen such that…”

Section: Methodsmentioning

confidence: 99%

Topological properties of a self-assembled electrical network via ab initio calculation

Stephenson

Lyon

Hübler

2017

Sci Rep

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Interacting electrical conductors self-assemble to form tree like networks in the presence of applied voltages or currents. Experiments have shown that the degree distribution of the steady state networks are identical over a wide range of network sizes. In this work we develop a new model of the self-assembly process starting from the underlying physical interaction between conductors. In agreement with experimental results we find that for steady state networks, our model predicts that the fraction of endpoints is a constant of 0.252, and the fraction of branch points is 0.237. We find that our model predicts that these scaling properties also hold for the network during the approach to the steady state as well. In addition, we also reproduce the experimental distribution of nodes with a given Strahler number for all steady state networks studied.

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“…However, for S 1 we only have outer part § , while for S N +1 we only have an inner part. Nevertheless, we still preserve the inner and outer notation for S 1 and S N +1 by writing A set of conductors in succesive embedding has some particular properties (see appendix C in Ref [4]). First, the only non-zero coefficients of capacitance are the diagonal ones C ii and the terms of the form C i,i±1 .…”

Section: A2 Electric Fields and Charge Densities For A Prolate Sphermentioning

confidence: 99%

Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries

Morales

Díaz

Herrera

2015

Journal of Electrostatics

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We find solutions of Laplace's equation with specific boundary conditions (in which such solutions take either the value zero or unity in each surface) using a generic curvilinear system of coordinates. Such purely geometrical solutions (that we shall call Basic Harmonic Functions BHF's) are utilized to obtain a more general class of solutions for Laplace's equation, in which the functions take arbitrary constant values on the boundaries. On the other hand, the BHF's are also used to obtain the capacitance of many electrostatic configurations of conductors. This method of finding solutions of Laplace's equation and capacitances with multiple symmetries is particularly simple, owing to the fact that the method of separation of variables becomes much simpler under the boundary conditions that lead to the BHF's. Examples of application in complex symmetries are given. Then, configurations of succesive embedding of conductors are also examined. In addition, expressions for electric fields between two conductors and charge densities on their surfaces are obtained in terms of generalized curvilinear coordinates. It worths remarking that it is plausible to extrapolate the present method to other linear homogeneous differential equations.

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The positivity and other properties of the matrix of capacitance: Physical and mathematical implications

Cited by 15 publications

References 26 publications

Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors

Robustness of subwavelength devices: a case study of cochlea-inspired rainbow sensors

Topological properties of a self-assembled electrical network via ab initio calculation

Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries

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