The elusive half-pole in the frequency domain transfer function of Peltier thermoelectric devices

A half-pole can be expected in the transfer function of a Peltier device because proportionality between the diffusion length and the square root of the diffusion time is intrinsic in the diffusion equation. The resulting -1∕2 bilogarithmic slope (10 dB∕dec) is, however, easily masked by the thermal time constant of the load, which makes it elusive. The goal of this work is to identify the arrangements which can reveal and make usable the half-pole, because the latter can be instrumental in a servo control to increase the open-loop gain without risking instability. The diffusion equation was solved in a sine wave regime for a one-dimensional model of a Peltier device. The Laplace transform method was used, and the periodic solution was obtained using Cauchy's theorem and the method of residues. The -1∕2 slope of the half-pole appeared observable in a frequency range which can be several decades wide, depending on details of device configuration and considered position within. Amplitude and phase of temperature and heat flux in various spots are discussed with emphasis on the physical meaning, and a comparison is provided with solutions yielded by the lumped model, which cannot show the half-pole. An experimental check of the theoretical approach and analysis was made taking into account the deviations from one-dimensionality occurring in a real Peltier device. Given a constant amplitude sine wave injected current, the quadrature component of the Seebeck voltage across the whole series of junctions was identified as the most easily measurable quantity related to the thermal response of the device. Experimental results for the latter turned out in good agreement with analytical solutions.