Abstract: Considerable progress has been made recently in the development of techniques to determine exactly two-point resistances in networks of various topologies. In particular, a general resistance formula of a non-regular m×n resistor network with an arbitrary boundary is determined by the recursion-transform (RT) method. However, research on the complex impedance network is more difficult than that on the resistor network, and it is a problem worthy of study since the equivalent impedance has many different properties from equivalent resistance. In this study, the equivalent impedance of a non-regular m×n RLC network with an arbitrary boundary is studied based on the resistance formula, and the oscillation characteristics and resonance properties of the equivalent impedance are discovered. In the RLC network, it is found that our formula leads to the occurrence of resonances at the boundary condition holding a series of specific values with an external alternating current source. This curious result suggests the possibility of practical applications of our formula to resonant circuits.

含有任意边界的m×n阶RLC网络的等效复阻抗特性

[1]Asad, J.H., 2013a. Exact evaluation of the resistance in an infinite face-centered cubic network. J. Stat. Phys., 150(6):1177-1182.

[2]Asad, J.H., 2013b. Infinite simple 3D cubic network of identical capacitors. Mod. Phys. Lett. B, 27(15):1350112.

[3]Asad, J.H., Diab, A.A., Hijjawi, R.S., et al., 2013. Infinite face-centered-cubic network of identical resistors: application to lattice Green’s function. Eur. Phys. J. Plus, 128:2.

[4]Bao, A., Tao, H.S., Liu, H.D., et al., 2014. Quantum magnetic phase transition in square-octagon lattice. Sci. Rep., 4:6918.

[5]Baule, A., Mari, R., Bo, L., et al., 2013. Mean-field theory of random close packings of axisymmetric particles. Nat. Commun., 4:2194.

[6]Bianco, B., Giordano, S., 2003. Electrical characterization of linear and non-linear random networks and mixtures. Int. J. Circ. Theory Appl., 31(2):199-218.

[7]Chair, N., 2012. Exact two-point resistance, and the simple random walk on the complete graph minus N edges. Ann. Phys., 327(12):3116-3129.

[8]Chair, N., 2014a. The effective resistance of the N-cycle graph with four nearest neighbors. J. Stat. Phys., 154(4):1177-1190.

[9]Chair, N., 2014b. Trigonometrical sums connected with the chiral Potts model, Verlinde dimension formula, two-dimensional resistor network, and number theory. Ann. Phys., 341:56-76.

[10]Chamberlin, R.V., 2000. Mean-field cluster model for the critical behaviour of ferromagnets. Nature, 408:337-339.

[11]Chitra, R., Kotliar, G., 2000. Dynamical mean-field theory and electronic structure calculations. Phys. Rev. B, 62:12715.

[12]Cserti, J., 2000. Application of the lattice Green’s function for calculating the resistance of an infinite network of resistors. Am. J. Phys., 68(10):896-906.

[13]Essam, J.W., Tan, Z.Z., Wu, F.Y., 2014. Resistance between two nodes in general position on an m×n fan network. Phys. Rev. E, 90(3):032130.

[14]Essam, J.W., Izmailyan, N.S., Kenna, R., et al., 2015. Comparison of methods to determine point-to-point resistance in nearly rectangular networks with application to a ‘hammock’ network. R. Soc. Open Sci., 2:140420.

[15]Gabelli, J., Fève, G., Berroir, J.M., et al., 2006. Violation of Kirchhoff’s laws for a coherent RC circuit. Science, 313(5786):499-502.

[16]Georges, A., Kotliar, G., Krauth, W., et al., 1996. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys., 68(1):13-125.

[17]Giordano, S., 2005. Disordered lattice networks: general theory and simulations. Int. J. Circ. Theory Appl., 33(6): 519-540.

[18]Haule, K., 2007. Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base. Phys. Rev. B, 75(15):155113.

[19]Izmailian, N.S., Huang, M.C., 2010. Asymptotic expansion for the resistance between two maximum separated nodes on an M by N resistor network. Phys. Rev. E, 82(1):011125.

[20]Izmailian, N.S., Kenna, R., 2014. A generalised formulation of the Laplacian approach to resistor networks. J. Stat. Mech. Theory Exp., 2014(9):09016.

[21]Izmailian, N.S., Kenna, R., Wu, F.Y., 2014. The two-point resistance of a resistor network: a new formulation and application to the cobweb network. J. Phys. A, 47(3): 035003.

[22]Kirchhoff, G., 1847. Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys., 148(12):497-508 (in German).

[23]Kirkpatrick, S., 1973. Percolation and conduction. Rev. Mod. Phys., 45(4):574-588.

[24]Klein, D.J., Randić, M., 1993. Resistance distance. J. Math. Chem., 12(1):81-95.

[25]Tan, Z.Z., 2011. Resistance Network Moder. Xidian University Press, Xi’an, China, p.28-146 (in Chinese).

[26]Tan, Z.Z., 2015a. Recursion-transform approach to compute the resistance of a resistor network with an arbitrary boundary. Chin. Phys. B, 24(2):020503.

[27]Tan, Z.Z., 2015b. Recursion-transform method for computing resistance of the complex resistor network with three arbitrary boundaries. Phys. Rev. E, 91(5):052122.

[28]Tan, Z.Z., 2015c. Recursion-transform method to a non-regular m×n cobweb with an arbitrary longitude. Sci. Rep., 5:11266.

[29]Tan, Z.Z., 2017. Two-point resistance of a non-regular cylindrical network with a zero resistor axis and two arbitrary boundaries. Commun. Theor. Phys., 67(3):280-288.

[30]Tan, Z.Z., Fang, J.H., 2015. Two-point resistance of a cobweb network with a 2r boundary. Commun. Theor. Phys., 63(1):36-44.

[31]Tan, Z.Z., Zhang, Q.H., 2015. Formulae of resistance between two corner nodes on a common edge of the m×n rectangular network. Int. J. Circ. Theory Appl., 43(7):944-958.

[32]Tan, Z.Z., Zhang, Q.H., 2017. Calculation of the equivalent resistance and impedance of the cylindrical network based on recursion-transform method. Acta Phys. Sin., 66(7):070501 (in Chinese).

[33]Tan, Z.Z., Zhou, L., Yang, J.H., 2013. The equivalent resistance of a 3×n cobweb network and its conjecture of an m×n cobweb network. J. Phys. A, 46:195202.

[34]Tan, Z.Z., Essam, J.W., Wu, F.Y., 2014. Two-point resistance of a resistor network embedded on a globe. Phys. Rev. E, 90(1):012130.

[35]Tzeng, W.J., Wu, F.Y., 2006. Theory of impedance networks: the two-point impedance and LC resonances. J. Phys. A: Math. Gen., 39(27):8579-8591.

[36]Whan, C.B., Lobb, C.J., 1996. Complex dynamical behavior in RCL shunted Josephson tunnel junctions. Phys. Rev. E, 53(1):405-413.

[37]Wu, F.Y., 2004. Theory of resistor networks: the two-point resistance. J. Phys. A, 37(26):6653-6673.

[38]Xiao, W.J., Gutman, I., 2003. Resistance distance and Laplacian spectrum. Theor. Chem. Acc., 110(4):284-289.

[39]Zhou, L., Tan, Z.Z., Zhang, Q.H., 2017. A fractional-order multifunctional n-step honeycomb RLC circuit network. Front. Inform. Technol. Electron. Eng., 18(8):1186-1196.