A Time-domain Approach to the 3-omega Heat Conductivity Measurement Method

The so-called “3ω method” is a well established method to measure heat conductivity of solids. It is a frequency-based method, in which the ratio between the first and third harmonics of an induced voltage in an electric heater element can be shown to be (inversely) proportional to the thermal conductivity of a solid that the heater is in direct contact with. Commonly, the method utilizes Discrete Fourier analysis in an off-line setting, which is of course perfectly valid when measuring material properties in a static setting.
In this paper we propose to make use of the measurement principle in a dynamic setting. We propose a novel time-domain approach to 3ω measurement, which can easily be implemented in a cheap micro-controller due to its modest memory and sampling rate requirements, and therefore likely to be useful for feedback in control loops or similar applications.
The approach comprises two main elements, a discrete-time signal generator, which provides a steady-state sinusoidal current output, and a standard Luenberger-style state observer designed to estimate the associated third harmonic in the presence of noisy voltage measurements. We prove that the signal generator is robust to numerical inaccuracies. The approach is tested in simulation and on actual laboratory data, showing good agreement with traditional off-line analysis.