On The Equivalent Impedance of Two-Impedance Self-Similar Ladder Networks

Recurrent ladder network models are encountered in the analysis of many natural systems including wave propagation in transmission lines, porous electrode/electrolyte interfaces and viscoelastic (bio)materials. Here we re-visit the equivalent impedance of self-similar ladder networks composed of two arbitrary impedances recursively arranged, with the goal of correlating the behavior of the whole versus that of the part. We show that the macroscopic and microscopic scales are related via a generally complex-valued parametric scaling factor. We report on two new universal complex numbers that we denote as the Complex Golden Ratios . We also show that a lumped inductor (with its internal series resistance) or a lumped capacitor (with its parallel parasitic resistance) are actually equivalent to an infinite ladder network that involves resistors and imaginary resistors only. Finally, we show that the Cole-Davidson model, widely used in material characterization, naturally arises from the self-similar circuit ladder structure. Comparison with tree networks is also discussed.