A practical first step in Kramers–Kronig analysis for undergraduates using an RC circuit

Kramers–Kronig (KK) relations are applied to the frequency response of a simple series–parallel RC circuit. By measuring the amplitude of the current as a function of frequency, it is shown how to numerically retrieve the phase using KK transformations, and, conversely, measured phase data is used to reconstruct the amplitude. The consistency of the KK procedure is verified by recovering the current magnitude from the calculated phase, and vice versa. The frequency dependent impedance of the circuit is obtained from the current magnitude and the phase. This experiment and subsequent KK analysis reinforce foundational topics in physics and mathematics and also introduce undergraduate students to a technique used in spectroscopy and material characterization. The computer programs necessary to perform the KK transformations are included.