Construções consistentes de espaços de Banach C (K) com poucos operadores

Immittance Data Validation by Kramers‐Kronig Relations – Derivation and Implications

“…Most important, we use below in (11), (12) and (13) which respectively represent time domain linearity, stability and causality, the fact that (7)…”

“…More generally, immittance could also include dielectric quantities such as complex permittivity (dielectric constant), e, susceptibility, c and modulus, M as well as transfer functions based on stimuli (excitation) other than electrical. [1][2][3] But prior to a meaningful analysis of conventional immittance measurements and likewise of similar complex valued quantities (absorptance, transmittance, reflectance, complex compliance, relaxation modulus) in many other fields of natural sciences and engineering, [7][8][9][10][11][12][13][14][15][16][17] the acquired data have to conform to the very principles of linear, time invariant (LTI) systems: linearity, stability, boundedness, continuity (consistency), uniqueness and causality. [1][2][3][4][5] This may either be checked experimentally or numerically, for example, using fast Fourier transformation (FFT) [18][19][20][21] on Kramers-Kronig (KK) relations [22][23][24] in the positive frequency domain (w !…”

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

“…The first principle derivation of linear dispersion relations truly applicable to irrational immittances (transfer functions) occurring in many other fields of natural sciences and engineering [2][3][4][5][6][7][8][9][13][14][15][16][17] is a future challenge. Note, spatial coordinate, wave number, momentum and energy could readily replace w and n as variables in the relations (1), (6) and (13) when applied to complex valued quantities other than immittances.…”

“…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, C = (jw) À1 Y, complex inductance, L = (jw) À1 Z) 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.…”

“…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, C = (jw) À1 Y, complex inductance, L = (jw) À1 Z) 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.…”