Dispersionless charge transfer on lumped-element transmission lines synthesized via resonant frequency assignment

Abstract

The uniform lumped element transmission line that exhibits translational symmetry is an inherently dispersive network. The dispersion properties can be transformed by imposing a harmonic symmetry that requires all of the resonant frequencies of the network to be integral multiples of some fundamental frequency. The resulting line will have spatially varying component values, thus breaking the network's translational symmetry. If mirror symmetry is imposed on the line such that right and left sides of the line have identical components, the natural resonant modes of the line will then be either spatially even or spatially odd. The even harmonic frequencies may be assigned to the spatially even modes, and the odd harmonic frequencies may be assigned to the spatially odd modes by methods similar to those used for Marx networks [1], [2], [3], [4]. The resulting network can then be shown to have several important properties: (a) All voltages and currents will be periodic in time with the fundamental period. (b) Starting with zero initial currents and finite charge voltages, the mirror image of the initial conditions on each side of the line will appear on the opposite sides at odd multiples of the half period. (c) A similar result can be stated when starting with zero initial voltages and finite initial currents.While many solutions can be found, the smoothly varying regular solution is of special interest. It possesses the property of having relatively spatially uniform impedance with a cutoff frequency that decreases away from the line center. Thus, higher frequency waves, which travel more slowly, are reflected further from the ends of the line, such that any narrow-band wave packet is reflected periodically without dispersing.This network may be used for efficiently transporting capacitive stored energy on ordinary transmission lines. The lumped-element line is cut at the symmetry plane, and an arbitrary length of conventional transmission line of appropriate impedance is inserted in the gap. Discharging the end capacitor into the resulting network generates a Gaussian-like pulse on the transmission line that is later converted to the identical capacitive charge on the network at the other end of the line.