On the perturbation of a uniform tiling with resistors

2016 ◽  
Vol 30 (24) ◽  
pp. 1650166 ◽  
Author(s):  
M. Q. Owaidat ◽  
J. H. Asad ◽  
Zhi-Zhong Tan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Square Lattice ◽  
Resistor Network ◽  
Theoretical Results

The perturbation of a uniformly tiled resistor network by adding an edge (a resistor) to the network is considered. The two-point resistance on the perturbed tiling in terms of that on the perfect tiling is obtained using Green’s function. Some theoretical results are presented for an infinite modified square lattice. These results are confirmed experimentally by constructing an actual resistor lattice of size 13 × 13.

scholarly journals Localization for a line defect in an infinite square lattice

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120579 ◽  
Author(s):  
D. J. Colquitt ◽  
M. J. Nieves ◽  
I. S. Jones ◽  
A. B. Movchan ◽  
N. V. Movchan
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Asymptotic Expansions ◽  
Square Lattice ◽  
Line Defect ◽  
Linear Superposition ◽  
Asymptotic Expressions ◽  
Yield Information ◽  
Finite Line ◽  
Single Mass

Localized defect modes generated by a finite line defect composed of several masses, embedded in an infinite square cell lattice, are analysed using the linear superposition of Green's function for a single mass defect. Several representations of the lattice Green's function are presented and discussed. The problem is reduced to an eigenvalue system and the properties of the corresponding matrix are examined in detail to yield information regarding the number of symmetric and skew-symmetric modes. Asymptotic expansions in the far field, associated with long wavelength homogenization, are presented. Asymptotic expressions for Green's function in the vicinity of the band edge are also discussed. Several examples are presented where eigenfrequencies linked to this system and the corresponding eigenmodes are computed for various defects and compared with the asymptotic expansions. The case of an infinite defect is also considered and an explicit dispersion relation is obtained. For the case when the number of masses within the line defect is large, it is shown that the range of the eigenfrequencies can be predicted using the dispersion diagram for the infinite chain.

SUBSTITUTIONAL SINGLE RESISTOR IN AN INFINITE SQUARE LATTICE APPLICATION TO LATTICE GREEN'S FUNCTION

2010 ◽  
Vol 24 (19) ◽  
pp. 2057-2068 ◽  
Author(s):  
M. Q. OWAIDAT ◽  
R. S. HIJJAWI ◽  
J. M. KHALIFEH
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Experimental Results ◽  
Square Lattice ◽  
Perfect Lattice ◽  
Arbitrary Lattice ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

The resistance between two arbitrary lattice sites in an infinite square lattice of identical resistors is studied when the lattice is perturbed by substituting a single resistor using lattice Green's function. The relation between the resistance and the lattice Green's function for the perturbed lattice is derived. Solving Dyson's equation, the Green's function and the resistance of the perturbed lattice are expressed in terms of those of the perfect lattice. Numerical and experimental results are presented.

PERTURBATION OF AN INFINITE NETWORK OF IDENTICAL CAPACITORS

2007 ◽  
Vol 21 (02) ◽  
pp. 199-209 ◽  
Author(s):  
R. S. HIJJAWI ◽  
J. H. ASAD ◽  
A. J. SAKAJI ◽  
J. M. KHALIFEH
Keyword(s):  
Asymptotic Behavior ◽  
Green’S Function ◽  
Green's Function ◽  
Numerical Results ◽  
Square Lattice ◽  
Infinite Network ◽  
Arbitrary Lattice ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

The capacitance between any two arbitrary lattice sites in an infinite square lattice is studied when one bond is removed (i.e. perturbed). A connection is made between the capacitance and the lattice Green's function of the perturbed network, where they are expressed in terms of those of the perfect network. The asymptotic behavior of the perturbed capacitance is investigated as the separation between the two sites goes to infinity. Finally, numerical results are obtained along different directions and a comparison is made with the perfect capacitances.

INFINITE NETWORK OF IDENTICAL CAPACITORS BY GREEN'S FUNCTION

2005 ◽  
Vol 19 (24) ◽  
pp. 3713-3721 ◽  
Author(s):  
J. H. ASAD ◽  
R. S. HIJJAWI ◽  
A. J. SAKAJI ◽  
J. M. KHALIFEH
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Lattice Site ◽  
Linear Chain ◽  
Tight Binding ◽  
Square Lattice ◽  
Van Hove Singularity ◽  
Infinite Network ◽  
Infinite Networks ◽  
Infinite Linear

The capacitance between arbitrary nodes in perfect infinite networks of identical capacitors is studied. We calculate the capacitance between the origin and the lattice site (l, m) for an infinite linear chain, and for an infinite square network consisting of identical capacitors using the Lattice Green's Function. The asymptotic behavior of the capacitance for an infinite square lattice is investigated for infinite separation between the origin and the site (l, m). We point out the relation between the capacitance of the lattice and the van Hove singularity of the tight-binding Hamiltonian. This method can be applied directly to other lattice structures.

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