Using a Natural Ratio to Compare DC and AC Resistances

Abstract

The simulation and construction of a direct current (DC) and alternating current (AC) resistor, based on a silicon wafer, has been described and demonstrated. By applying the van der Pauw method and the Thompson-Lampard theorem, to within approximations accommodating the conditions of the resistor's construction, a constant resistance ratio, (π/ln2)2, was derived that is independent of the sample resistivity and thickness. The constant ratio, valued at approximately 20.5, can theoretically be used as a basis of comparison between two distinct calibration chains, one based on the traceability from a calculable capacitor and the other based on the quantum Hall effect. To support the calculated ratio, several sets of simulations were performed for both DC and AC cases. The DC simulation results agreed with the ratio value to within 0.035 % when using a wafer thickness of 0.53 mm. Additionally, the experimental DC and AC (1 kHz) results agreed with the calculated ratio value to within 0.23 %, with at most a 0.06 % standard uncertainty before point contact errors from device fabrication.