ABSTRACT
Exact data of an electric circuit (EC) model of RLC (resistor, inductor, capacitor) elements representing rational immittance of LTI (linear, time invariant) systems are numerically Fourier transformed to demonstrate within error bounds applicability of the Hilbert integral tranform (HT) and Kramers-Kronig (KK) integral tranform (KKT) method. Immittance spectroscopy (IS) data are validated for their HT (KKT) compliance using non-equispaced fast Fourier transformation (NFFT) computations. Failing of HT (KKT) testing may not only stem from non-compliance with causality, stability and linearity which are readily distinguished using anti HT (KKT) relations. It could also indicate violation of uniform boundedness to be overcome either by using singly or multiply subtracted KK transform (SSKK or MSKK) or by seeking KKT of the same set of data at a complementary immit- tance level. Experimental IS data of a fuel cell (FC) are also numerically HT (KKT) validated by NFFT assessing whether LTI principles are met. Figures of merit are suggested to measure success in numerical validation of IS data.
ABSTRACT
Explicitly based on causality, linearity (superposition) and stability (time invariance) and implicit on continuity (consistency), finiteness (convergence) and uniqueness (single valuedness) in the time domain, Kramers-Kronig (KK) integral transform (KKT) relations for immittances are derived as pure mathematical constructs in the complex frequency domain using the two-sided (bilateral) Laplace integral transform (LT) reduced to the Fourier domain for sufficiently rapid exponential decaying, bounded immittances. Novel anti KK relations are also derived to distinguish LTI (linear, time invariant) systems from non-linear, unstable and acausal systems. All relations can be used to test KK transformability on the LTI principles of linearity, stability and causality of measured and model data by Fourier transform (FT) in immittance spectroscopy (IS). Also, integral transform relations are provided to estimate (conjugate) immittances at zero and infinite frequency particularly useful to normalise data and compare data. Also, important implications for IS are presented and suggestions for consistent data analysis are made which generally apply likewise to complex valued quantities in many fields of engineering and natural sciences.
