Mathematical solutions of comprehensive variations of a transmission-line model of the theoretical impedance of porous electrodes

•Mathematical solutions for impedance of several types of transmission-line model were obtained.•A comprehensive set of specific solutions with possible all boundary conditions is presented.•An equivalent transform from a transmission-line model into a Y-circuit is proposed.

Mathematical solutions were obtained for the impedance of one-dimensional transmission-line (TML) models with all possible variations of the boundary conditions and connections to an external circuit. First, three types of connection (Z-type, T-type, and E-type) were considered. A more generalized connection that includes the characteristics of both Z-type and T-type (universal-type) was also considered. All four of the resulting general solutions were versatile, but quite complicated. Next, comprehensive variations of the general solutions depending on specific boundary conditions, which are generally simpler, were also considered. Finally, an equivalent transform from a three-terminal TML model to a simple Y-circuit was calculated. This enables calculation of the serial connection of multiple TML models with different parameters, which represents a multi-layered electrode.