Perturbation of an infinite network of identical capacitors
20 Jan 2007-International Journal of Modern Physics B (World Scientific Publishing Company)-Vol. 21, Iss: 02, pp 199-209
TL;DR: In this paper, the capacitance between any two arbitrary lattice sites in an infinite square lattice is studied when one bond is removed (i.e. perturbed), and a connection is made between capacitance and the lattice Green's function of the perturbed network, where they are expressed in terms of those of the perfect network.
Abstract: 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.
Citations
More filters
TL;DR: It is found that the formulation of the equivalent resistance for the m × n resistor network leads to the occurrence of resonances at frequencies associated with (n + 1)ϕt = kπ, which suggests the possibility of practical applications of the formulae to resonant circuits.
Abstract: Summary
This paper deals with the equivalent resistance for the m × n resistor network in both finite and infinite cases. Firstly, we build a difference equation driven by a tridiagonal matrix to model the network; then by performing the diagonalizing transformation on the driving matrix, and using the auxiliary function tz(x,n), we derive two formulae of the equivalent resistance between two corner nodes on a common edge of the network. By comparing two different formulae, we also obtain a new trigonometric identity here. Our framework can be effectively applied in complex impedance networks. As in applications in the LC network, we find that our formulation leads to the occurrence of resonances at frequencies associated with (n + 1)ϕt = kπ. This somewhat curious result suggests the possibility of practical applications of our formulae to resonant circuits. At the end of the paper, two other formulae of an m × n resistor network are proposed. Copyright © 2014 John Wiley & Sons, Ltd.
33 citations
TL;DR: In this paper, the effective resistance between two arbitrary nodes of a resistor network is investigated when it is perturbed by introducing two extra interstitial resistors in the perfect lattice using lattice Green's function.
Abstract: Investigation of the effective resistance between two arbitrary nodes of a resistor network is carried out when it is perturbed by introducing two extra interstitial resistors in the perfect lattice using lattice Green’s function. Analytical formula for the effective resistance of the perturbed lattice is given in terms of that for the unperturbed one. Numerical results for a square lattice are presented.
29 citations
TL;DR: New fundamentals of the 2 × n RLC circuit network in the fractional-order domain are introduced and the new phenomena and laws are presented by the results of the numerical simulations, which are impossible in the conventional cases.
Abstract: This paper introduces new fundamentals of the 2 × n RLC circuit network in the fractional-order domain. First, we derive the three general formulae of the equivalent impedances of the circuit network by using the matrix transform methods and constructing the differential equation models in three different cases. Moreover, we systematically study the effects of the system parameters on the impedence characteristics in the three different cases. Specifically, the new phenomena and laws are presented by the results of the numerical simulations, which are impossible in the conventional cases. Finally, a comparative sensitivity analysis about the three cases with respect to the fractional orders for the fractional-order circuit network is carried out in detail. Mathematical analyses and numerical simulations are included to validate the study.
23 citations
TL;DR: A quaternion matrix equation is built and the method of matrix transformations in terms of the network analysis is proposed and found that the equivalent resistance is expressed by coskπ/9 in a series of strict calculation.
Abstract: A classic problem in electric circuit theory studied by numerous authors over 160 years is the computation of the resistance between two nodes in a resistor network, yet some basic problem in m×n cobweb network is still not solved ideally. The equivalent resistance and capacitance of 4×n cobweb network are investigated in this paper. We built a quaternion matrix equation and proposed the method of matrix transformations in terms of the network analysis. We proposed a brief equivalent resistance formula and find that the equivalent resistance is expressed by coskπ/9 in a series of strict calculation. Meanwhile, an equivalent resistance of infinite networks is gained. Using the inverse mapping relation between capacitance parameters and resistance parameters, the equivalent capacitance formula is also given for the 4×n capacitance cobweb network. By analyzing and comparing the equivalent resistances of the 1×n, 2×n, 3×n and 4×n cobweb networks, two conjectures on the equivalent resistance and capacitance of the m×n cobweb network are proposed. Copyright © 2013 John Wiley & Sons, Ltd.
23 citations
TL;DR: In this paper, the capacitance between arbitrary two sites (vertices) in infinite triangular and honeycomb networks is studied by using Green's function, and the results show that the capacity of the honeycomb lattice is the same as that of the triangular lattice.
Abstract: The capacitance between arbitrary two sites (vertices) in infinite triangular and honeycomb networks is studied by using Green’s function. Recurrence formulas for capacitance between arbitrary sites of the triangular lattice are obtained. The capacitance for the honeycomb lattice is shown to be expressed in terms of the one for the triangular lattice.
17 citations
References
More filters
Book•
01 Jan 1950
TL;DR: In this paper, the Fourier integral is used as the basis of the operational calculus, and the convergence of the definition integral is discussed. But it is not discussed in detail.
Abstract: General introduction The Fourier integral as basis of the operational calculus Elementary operational images Elementary rules The delta or impulse function Questions concerning the convergence of the definition integral Asymptotic relations and operational transposition of series Linear differential equations with constant coefficients Simultaneous linear differential equations with constant coefficients Electric-circuit theory Linear differential equations with variable coefficients Operational rules of more complicated character Step functions and other discontinuous functions Difference equations Integral equations Partial differential equations in the operational calculus of one variable Simultaneous operational calculus 'Grammar' 'Dictionary' List of authors quoted General index.
328 citations
TL;DR: In this article, the resistance between two arbitrary grid points of several infinite lattice structures of resistors is calculated by using lattice Green's functions and the relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian is given.
Abstract: The resistance between two arbitrary grid points of several infinite lattice structures of resistors is calculated by using lattice Green’s functions. The resistance for d dimensional hypercubic, rectangular, triangular, and honeycomb lattices of resistors is discussed in detail. Recurrence formulas for the resistance between arbitrary lattice points of the square lattice are given. For large separation between nodes the asymptotic form of the resistance for a square lattice and the finite limiting value of the resistance for a simple cubic lattice are calculated. The relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian is given. The Green’s function method used in this paper can be applied in a straightforward manner to other types of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.
259 citations
TL;DR: In this paper, the resistance between two arbitrary grid points of several infinite lattice structures of resistors was calculated by using lattice Green's functions, and the relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian was shown.
Abstract: We calculate the resistance between two arbitrary grid points of several infinite lattice structures of resistors by using lattice Green's functions. The resistance for $d$ dimensional hypercubic, rectangular, triangular and honeycomb lattices of resistors is discussed in detail. We give recurrence formulas for the resistance between arbitrary lattice points of the square lattice. For large separation between nodes we calculate the asymptotic form of the resistance for a square lattice and the finite limiting value of the resistance for a simple cubic lattice. We point out the relation between the resistance of the lattice and the van Hove singularity of the tight-binding Hamiltonian. Our Green's function method can be applied in a straightforward manner to other types of lattice structures and can be useful didactically for introducing many concepts used in condensed matter physics.
181 citations
TL;DR: In this paper, the resistance between two arbitrary nodes in an infinite square lattice of identical resistors is calculated for infinite triangular and hexagonal lattices in two dimensions, and also for infinite cubic and hypercubic lattice in three and more dimensions.
Abstract: The resistance between two arbitrary nodes in an infinite square lattice of identical resistors is calculated. The method is generalized to infinite triangular and hexagonal lattices in two dimensions, and also to infinite cubic and hypercubic lattices in three and more dimensions.
109 citations
TL;DR: In this article, the authors present a complete elliptic integral of the first kind with complex modulus; the integral has been found to be evaluated efficiently by the method of the arithmetic-geometric mean.
Abstract: Formulas are provided which are convenient for the evaluation of the lattice Green's functions for the bcc, fcc, and rectangular lattices, at an arbitrary complex variable. The formulas involve the complete elliptic integral of the first kind with complex modulus; the integral has been found to be evaluated efficiently by the method of the arithmetic‐geometric mean, generalized for the case with complex modulus. The expansions of the lattice Green's functions around the singular points are given for the bcc and fcc lattices. These lattice Green's functions diverge at a variable. The singular points responsible for the divergences are found to form one‐dimensional lines.
80 citations