Graphical Conversion between Compliance and Modulus, Permittivity and Electric Modulus, and Impedance and Admittance

Spectrometries probing relaxation and retardation phenomena, such as dielectric, mechanical, and impedance spectroscopies, often require the analyses with both susceptibilities spectra and its reciprocals (e.g., complex permittivity and electric modulus, mechanical compliance and mechanical modulus, and impedance and admittance). In the present paper, the geometric relation between and is derived and the procedure to convert into on a Cole-Cole diagram is proposed. This method helps us to relate them intuitively and yields clearer understanding on their interrelations. Moreover, it opens the new route for the geometric approach to derive many mathematical properties of spectra. The relation between peak position of spectrum and that of spectrum and the shape of spectra are discussed on the basis of this method.