Two-point resistance of an m×n resistor network with an arbitrary boundary and its application in RLC network

A rectangular m × n resistor network with an arbitrary boundary is investigated, and a general resistance formula between two nodes on an arbitrary axis is derived by the Recursion-Transform (RT) method, a problem that has never been resolved before, for the Green's function technique and the Laplacian matrix approach are inapplicable to it. To have the exact solution of resistance is important but it is difficult to obtain under the condition of arbitrary boundary. Our result is directly expressed in a single summation and mainly composed of characteristic roots, which contain both finite and infinite cases. Further, the current distribution is given explicitly as a byproduct of the method. Our framework can be effectively applied to RLC networks. As an application to the LC network, we find that our formulation leads to the occurrence of resonances at h 1 = 1 − cos i − sin i cot n i . This somewhat curious result suggests the possibility of practical applications of our formulae to resonant circuits.

[1]  Zhi-Zhong Tan,et al.  Recursion-Transform method to a non-regular m×n cobweb with an arbitrary longitude , 2015, Scientific Reports.

[2]  Keiichi Nakayama,et al.  A Series RCL Circuit Theory for Analyzing Non-Steady-State Water Uptake of Maize Plants , 2014, Scientific Reports.

[3]  B. Normand,et al.  Partial Order in Potts Models on the Generalized Decorated Square Lattice , 2013, 1304.7146.

[4]  J. Khalifeh,et al.  Perturbed infinite 3D simple cubic network of identical capacitors , 2013, 1302.4376.

[5]  Kou Su-peng,et al.  Quantum Fidelity and Thermal Phase Transitions in a Two-Dimensional Spin System , 2012 .

[6]  Zhi-Zhong Tan,et al.  Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  J. Cserti.  Application of the lattice Green's function for calculating the resistance of an infinite network of resistors , 1999, cond-mat/9909120.

[8]  Whan Cb,et al.  Complex dynamical behavior in RCL-shunted Josephson tunnel junctions. , 1996 .

[9]  I. Gutman,et al.  Resistance distance and Laplacian spectrum , 2003 .

[10]  Jihad H. Asad,et al.  Exact Evaluation of the Resistance in an Infinite Face-Centered Cubic Network , 2012, 1210.4725.

[11]  N.Sh. Izmailian,et al.  The two-point resistance of a resistor network: a new formulation and application to the cobweb network , 2013, 1310.1335.

[12]  Zhi-zhong Tan,et al.  The equivalent resistance of a 3 × n cobweb network and its conjecture of an m × n cobweb network , 2013 .

[13]  F. Y. Wu,et al.  Theory of impedance networks: the two-point impedance and LC resonances , 2006 .

[14]  Jia Li-ping,et al.  Propagation and Interaction of Edge Dislocation (Kink) in the Square Lattice , 2015 .

[15]  Zhi-Zhong Tan,et al.  Two-point resistance of a resistor network embedded on a globe. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  F. Y. Wu.  Theory of resistor networks: the two-point resistance , 2004 .

[17]  Zhi-Zhong Tan,et al.  Resistance between two nodes in general position on an m×n fan network. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Zhi-Zhong Tan,et al.  Two-Point Resistance of a Cobweb Network with a 2r Boundary , 2014, 1407.3693.

[19]  S. Kirkpatrick.  Percolation and Conduction , 1973 .