Evaluation of dielectric permittivity and conductivity by time domain spectroscopy. Mathematical analysis of Fellner-Feldegg's thin cell method

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Abstract
A mathematical analysis is presented of Fellner-Feldegg's thin cell method from which the permittivity and conductivity of dielectric materials can be obtained in the time domain. It is proved that the thin cell response has to be interpreted as a Taylor expansion of the step response around l=0. The first term in the expansion (Fellner-Feldegg's thin cell relation) can be found by integration over the ordinary Laplace contour (c − i ∞, c + i ∞). The other coefficients can be obtained from complex integration over a contour W enclosing all singularities of the integrand. Restrictions on the thickness of the sample are derived from the requirement that the second term in the expansion must be small relative to the first one. The step response is also calculated with asymptotic methods and with numerical methods for three currently used permittivity relations. Also, conductivity is included. For Debye materials a relation is found that tends to give a better fit to the exact response than the thin cell relation. For such materials the relaxation time found experimentally, using the thin cell relation, is always too large. An iteration procedure is suggested to obtain the correct relaxation time. For conductive materials the thin cell approximation leads to an incorrect value of the offset of the base line. The correct relation is derived.