Frequency response of unified dielectric and conductive systems involving an exponential distribution of activation energies

When the small-signal ac frequency response of a dielectric or conductive system is known, either functionally or as data, it is shown that the corresponding response of an associated conductive or dielectric system may be immediately obtained through the use of new duality relations. A specific model is considered which involves thermally activated capacitance and/or resistance, with an activation energy probability density exponentially dependent on energy. Previous frequency response analyses of such a continuously distributed model involve inadequate approximations and lead to erroneous predictions. Correct immittance results are presented in three ways: analytically, by means of complex plane plots, and through the use of three-dimensional perspective plots. Results are given in general form but apply to both dielectric and conductive systems which involve the same functional dependence on activation energies. Low- and high-frequency-limiting responses for a given system are found to be associated with the same simple equivalent circuit. In intermediate frequency ranges a power-law frequency response somewhat like that of the constant phase element may occur. Differences between the power-law exponents for dielectric and conductive systems are clarified, and the types of possible temperature dependence of the exponents explored. Exponent values are not limited to the range between zero and unity. The overall response of the present normalized three-parameter model is similar to that often found experimentally for both dielectric and conductive systems and similar to but more general than that of other normalized distributed-element (two-parameter) models such as that of Williams and Watts and that of Davidson and Cole.