Resistance and capacitance of 4 × n cobweb network and two conjectures

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 cos(kπ/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.     

Formulae of resistance between two corner nodes on a common edge of the m × n rectangular network

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)?_t = 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.     

Application of the recursion‐transform method in calculating the equivalent resistance of two‐dimensional finite resistor network

Looking for a handy and exact calculation for the equivalent resistance of an M × N resistor network is important but difficult, even for the rectangular resistor network. In this paper, we calculate the equivalent resistance of an M × N rectangular resistor network by means of the recursion‐transform method, where the idea of multiple external current sources based on the typical mesh current is used, and find a new formula of equivalent resistance which is different from the result of the other paper. In our scheme, recalculations are not required to obtain the equivalent resistance between different terminals. We further investigate how the order of resistor network and the ratio between two unit resistances affect the equivalent resistance. We find that the equivalent resistance between arbitrary terminals tends to a constant as the order and ratio increase when M is given. Copyright © 2016 John Wiley & Sons, Ltd.     

Experimental studies of optical Raman amplifiers for 1.3‐µm wavelength band and its application to 20 Gbit/s × 2 WDM transmission

Configurations of a Raman amplifier suitable for a 1.3‐µm wavelength band are discussed and their properties are experimentally investigated. Pump light with a wavelength of 1.23 µm that is necessary for the Raman amplification in the 1.3‐µm wavelength band is obtained using a 1.06‐µm fiber laser and Raman laser technique. Concerning the Raman laser, wavelength conversion from 1.06 µm to 1.23 µm is effectively achieved using a cavity configuration including fiber Bragg gratings and a dispersion‐shifted fiber. On the other hand, a conventional dispersion compensation fiber which has an essential property of high nonlinearity is applied in order to obtain large gain at 1.3 µm. Net gain of 35 dB and output power of 15 dBm are achieved. To confirm the applicability of the Raman amplifiers to high‐speed optical transmissions, experiments of 20 Gbit/s × 2 WDM repeaterless transmission through a 80‐km conventional single‐mode fiber are carried out. The 1.3‐µm signal should be degraded due to the dispersion caused by the dispersion compensation fiber in the Raman amplifier; However, bit error rate of less than 10 to 12 is obtained at both wavelengths, which is sufficient performance for practical uses. © 2003 Wiley Periodicals, Inc. Electr Eng Jpn, 143(1): 58–65, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.10137