Recursive Star-Mesh Transformations for Resistance Standards

A recent mathematical framework for optimizing resistor networks to achieve values in the MΩ through GΩ levels was employed for two specific cases. Objectives here include proof of concept and identification of possible apparatus limitations for future experiments involving graphene-based quantum Hall array resistance standards. Using fractal-like, or recursive, features of the framework allows one to calculate and implement network designs with substantially lower-valued resistors. The cases of 100 MΩ and 1 GΩ demonstrate that, theoretically, one would not need more than 100 quantum Hall elements to achieve these high resistances.