Immittance Data Validation using Fast Fourier Transformation (FFT) Computation – Synthetic and Experimental Examples

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 nonequispaced fast Fourier transformation (NFFT) computations. Failing of HT (KKT) testing may not only stem from noncompliance 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.In a companion communication, [1] we note the usefulness and need for data validation of immittance spectroscopy (IS) measurements and models. [2][3][4][5][6][7][8][9][10][11][12] Immittance (e. g. admittance, Y, impedance, Z, complex capacitance,is known under various other names and appears in modified form too in many other fields of natural sciences and engineering [6][7][8][9][10][11][12] where validation of measurements and verification of model data is often likewise required. They basically adhere all to the same principles, comply with the same relations and face the same dilemmas as presented here.For inertial systems (materials, interfaces or devices) whether oscillatory (dynamic) or at rest, we derived for linear, stable & causal systems [13][14][15][16][17] in the real angular frequency domain (Fourier space), <efÀjsg ¼ w; s ¼ s þ jw; 0 s 2 R; w ¼ 2pf 2 R; ðAEjÞ 2 ¼ À1 using integral transform properties (theorems) [2,18,19] for continuous, bounded (convergent), rational immittance, of finite degree, deg N deg D < 1 with zeros (roots of N), z i 2 C and poles (roots of D), p j 2 C À :¼ fs 2 C : <es < 0g Hilbert integral transform (HT) and Kramers-Kronig (KK) integral transform (KKT) relations, [1] (1) with 2a 2 + a + 1 = b; * denotes complex conjugation. Taking a = 0 thus b = 1, (2) with g = 0, g = À1 and g = +2 are anti HT (KK) relations complementary to (1) when respectively IS linearity (superposition), causality (cause precedes its effect) and stability (dilation, translation & rotation invariance) is violated. [1] is the linear HT [2,[20][21][22][23] (forward, H þ and inverse H À ) on R where P signifies the (Cauchy's) principal value taken at w = n and all I(w) poles (isolated singularities).is the linear KKT (forward, K þ and inverse K À ) pair on R þ :¼ fx 2 R : 0 xg. It is also known as dispersion relation [13][14][15][16][17][24][25][26] for the real and imaginary immittance parts, <eIðwÞ ¼ 0:5ðI þ I * ÞðwÞ and...

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