In the 3ω method a metal line on the sample carries current at frequency ω; its temperature oscillates at 2ω, and the resistance oscillation mixes a small voltage at 3ω. Over the linear-regime window the in-phase temperature amplitude falls as a straight line against ln f — and that slope is inversely proportional to the substrate’s thermal conductivity. Turn the conductivity dial and watch the slope flatten; add noise and see why the fit must span a real frequency decade.
Model: linear-regime 3ω response with in-phase amplitude ΔT = a − (b/k) ln f (normalized), an ILLUSTRATIVE idealized constant out-of-phase reference (real phase behavior varies with frequency, substrate, line width and model), independent Gaussian noise per point (fixed seed), ordinary least squares over the plotted window. Prefactors illustrative; a real reduction uses the full line-source solution with measured heater geometry. Schematic teaching tool, not an instrument.