Kronig-Kramers Relations Research Articles

Splitting Magnitude Response into Real and Imaginary Parts

The determination of real and imaginary parts from magnitude responses is studied for causal linear time-invariant systems having monotonic impulse responses. It is demonstrated that the problem can be interpreted as a special filtering task in the Mellin transform domain having a diffuse magnitude response bounded by the magnitude responses of the filters corresponding to zero and maximum imaginary parts prescribed by the Kronig-Kramers relations. Discrete-time filters processing geometrically sampled magnitude responses are designed for determining the real and imaginary parts. Testing results are presented verifying the performance of the filters.

Extension of Modified Havriliak-Negami Model to Characterize Linear Viscoelastic Properties of Asphalt Binders

AbstractAccurate characterization of mechanical properties of asphalt binders plays an important role in the evaluation and selection of binders, and in the design and analysis of mixtures. In this study the modified Havriliak-Negami model previously proposed for complex modulus of asphalt concrete is extended to account for the steady-state flow behavior of viscoelastic liquid materials. The analytical forms of storage and loss modulus master curves are derived from the complex modulus model through complex algebra. The Kronig-Kramers relations of the model are satisfied so that various LVE functions are equivalent and the causality of the model is fulfilled. In addition, the same time-temperature shift factor applies to various LVE functions. Thus, the model is consistent with the LVE theory. The model is fitted to complex modulus test results of various asphalt binders. The results show that the model can accurately characterize the properties of asphalt binders, regardless of whether they show viscoel...

Extraction of Dielectric Constant and Loss Tangent Using New Rapid Plane Solver and Analytical Debye Modeling for Printed Circuit Boards

Dielectric material properties of printed circuit boards (PCBs) are needed by designers working in various areas such as signal integrity, antennas, and embedded RF components. Among many methods to extract the material properties, the full sheet resonance technique is commonly used on PCBs due to its simplicity. The disadvantage of this method is that an analytical equation is used to extract the dielectric constant, which is accurate only for lossless dielectrics. In this paper, a new method is presented to solve the inaccuracy issue of the extraction of the dielectric constant by applying customized electromagnetic simulation based on a new rapid plane solver instead of analytical equations. For PCB dielectrics, the loss tangent tends to be flat over several decades. The dielectric constant then varies as a function of frequency based on the Kronig-Kramers relations. This paper introduces a new Debye type of a model for the complex permittivity of such dielectrics. The parameters of the Debye model can be obtained analytically without requiring any curve fitting. The resulting Debye model can then be easily integrated in SPICE or a finite-difference time-domain simulator.

Debye Model Fitting for Time-Domain Modeling of Lossy Dielectrics

In this paper, we will show a systematic procedure for fitting a Debye model to tabulated data of dielectric constant and loss tangent at discrete frequency points. Our approach is based on generation of a guaranteed passive RC macromodel, satisfying Kronig-Kramers relations for complex permittivity. We present for the first time an RC vector fitting method.

Optical Properties of DNA in Aqueous Solution

In the study of DNA electric birefringence, it is usual to use theories that consider that molecules in solution are small in relation to the light wavelength. In this work, we study the DNA electric birefringence using a broken-rod macroion (BRM) model composed of two cylindrical arms which does not restrict the size of the molecules. To achieve this, we include the inhomogeneity effect of the light electric field through the molecule and the interaction between its different parts. To analyze the interaction between a molecule and the incident beam of light, we apply the discrete dipole approximation (DDA), according to which each molecule is described as a finite array of electronic coupled oscillators. The electric birefringence is calculated from the oscillator polarizability. This is obtained from experimental data of electric birefringence saturation and from the increment of the solution refraction index in relation to that of the solvent. Furthermore, the oscillator polarizability is also estimated from DNA absorption spectrum using the Kronig-Kramers relations. This allows us to analyze the contributions of the different absorption bands of DNA to the electric birefringence. We analyze the influence of the inhomogeneity of the light electric field and of the intramolecular interactions in the characterization of DNA optical properties using electric birefringence measurements.

Nearly constant loss or constant loss in ionically conducting glasses: A physically realizable approach

Conductivity exhibiting power-law frequency response with an exponent of unity leads to frequency-independent dielectric loss. Such constant-loss (CL) behavior is not physically realizable over a nonzero frequency range, and approximate expressions that have been used to represent it are inconsistent with the Kronig–Kramers relations. Response models are proposed and investigated that do satisfy these relations and can lead to very close approximation to CL over many frequency decades, as often observed at low temperatures in ionic conductors such as glasses. Apparent CL response is shown to arise from the series connection of a constant-phase complex-power-law element (CPE), with exponent δ (0<δ≪1), and a frequency-independent dielectric constant, εU. Two physically disparate situations can lead to such a series connection. The first involves bulk CPE response in series with an electrode-related, double-layer blocking capacitance involving a dielectric constant εS. Then, apparent CL behavior may be associated with localized ionic motion in the bulk of the material. The second (mirror-image) situation involves CPE response associated with ionic motion in or at an electrode in series with a capacitance such as the bulk high-frequency-limiting total dielectric constant ε∞ or the pure-dielectric quantity εD∞. The present model is used to simultaneously fit both the real and imaginary parts of data derived from measurements on a sodium-trisilicate glass at 122 K. This data set exhibits power-law nearly constant loss for ε′(ω) and apparent CL for ε″(ω). The magnitude of the CL closely satisfies a simple equation involving only δ and εU. Further, for the electrode-power-law situation, estimated values of limiting-high-frequency dielectric constants turn out to be more consistent with bulk values established at much higher temperatures where nearly constant loss is no longer a dominant part of the response. Data at −0.5°C are also analyzed with a more complicated composite model, one that is a generalization of both of the above approaches, and nearly constant loss bulk, not electrode, power-law effects in both ε′(ω) and ε″(ω) are isolated and quantified. For this data set it is shown that electrode effects are important at both ends of the frequency range.

Alternatives to Kronig-Kramers transformation and testing, and estimation of distributions

Two alternatives to Kronig-Kramers analysis of small-signal ac immittance data are discussed and illustrated using both synthetic and experimental data. The first, a derivative method of approximating imaginary-part response from real-part data, is found to be too approximate in regions where the imaginary-part varies appreciably with frequency. The second, a distribution of relaxation-times fitting method, is shown to be valuable for testing whether a data set satisfies the Kronig-Kramers relations and so is associated with a system whose properties are time-invariant. It also is valuable for estimating real- or imaginary-part response from the other part, usually with small error. Unlike Kronig-Kramers analysis, the second method usually requires no extrapolation outside the range of the measured data. Finally, this discrete-function method also allows one to estimate the distribution of relaxation times or activation energies associated with a given set of frequency-response data. This application is described and illustrated for both synthetic and experimental data and is shown to yield good but somewhat approximate results for the estimation of continuous distributions. It is particularly valuable for identifying response regions arising from a continuous distribution and distinguishing them from those associated with discrete time-constant response.

Some new directions in impedance spectroscopy data analysis

New developments in two main data analysis areas are discussed: (a) complex nonlinear least squares (CNLS) fitting of data and (b) data-transforming and optimizing integral transforms. In the first category, a Monte Carlo study is used to answer the question of which of several different parameterizations of an ambiguous equivalent circuit model lead to minimum correlation between fitting parameters, a desirable condition. In addition, results are briefly discussed which address the questions of (1) what should be minimized in CNLS fitting? (2) how well can one discriminate between exact small-signal binary electrolyte response and conventional finite-length diffusion response? and (3) what is the ultimate precision of parameter estimates obtained in a CNLS fit? In the second area, new forms of the Kronig-Kramers relations (KKR) are discussed; the accuracy of several different ways of carrying out the numerical quadratures needed in such transforms is compared; and it is shown how random errors present in complex data are transformed by the KKR. Then, new transforms are described and illustrated that can replace exponential Fourier and KK transforms and, at the same time, can greatly reduce random error and some kinds of systematic errors in real, imaginary, or complex frequency response data or transient response data without the need for smoothing or filtering parameter choices.

The Calculation of Absolute Intensities Observed Due to Magneto-Optic Rotation in Atomic Vapors

The optical systems required to observe the Faraday or Voigt effects in atomic vapors are described in terms of the corresponding Jones matrices. The solution of the matrix equations reveals separate, additive dichroic and dispersive contributions to the transmitted intensity. The two contributions are combined by use of the Kronig-Kramers relations to give a final expression for the intensity due to magneto-optical rotation. This appears as the product of the intensity transmitted through the atomic vapor, the square of the zero field absorbance within the vapor, and the inverse square of a “Zeeman factor” z. Empirical data for the latter two parameters are currently available. Thus, the final formulae permit estimates of magneto-optic signal magnitudes to be made on the basis of presently existing information.

Applications of the Kronig-Kramers relations (and reply)

Aspects of the applications of the Kronig-Kramers relations in dielectric and viscoelastic theory and in electrical network analysis are discussed.

The failure of the modified Kronig-Kramers relations for second harmonic generation

Abstract Second harmonic generation can take place via magnetic resonance under saturation conditions. The modification of the Kronig-Kramers relations which applies (to the fundamental) under saturation condition is shown strictly not to apply in the second harmonic case.

The modification of the Kronig-Kramers relations under saturation conditions

Abstract When the real and imaginary parts of the susceptibility, χ′ and χ″ respectively, of narrow saturated absorption lines are of Lorentzian form, it is shown that a constant A can always be found such that χ′ and A χ″ are a Kronig-Kramers transform pair. Lines made up of weighted linear super-positions of Lorentzian lines having the same shape and the same saturation behaviour also possess this property, as do certain continuous distributions of narrow Lorentzian lines. The Kronig-Kramers calculation of a saturated absorption lineshape from a saturated dispersion signal is therefore very likely to yield the true lineshape.

A new use of the Kronig-Kramers relations in nuclear magnetic resonance. II

A new use of the Kronig-Kramers relations in nuclear magnetic resonance. II

Optical properties of copper and gold in the vacuum ultra-violet

The optical constants of copper and gold have been determined over a range of wavelengths from 4600 A to 460 A by measuring the reflection coefficient for unpolarized light at four angles of incidence. The results are expressed by means of graphs of 1 and 2, the real and imaginary parts of a complex dielectric constant, against photon energy, from 1 to 26 eV. They agree with recent determinations by Roberts and by Otto in the visible, but disagree somewhat with the work of Ehrenreich and Philipp and of Canfield, Hass and Hunter in the ultra-violet. Precautions were taken to avoid surface distortion and contamination and to correct for stray light, and it is claimed that on this account the present data, where they depart from the old, are to be preferred. The Kronig-Kramers relations are used to show that the data are internally consistent and also to suggest how they may be extrapolated to higher and lower energies.

A new use of the kronig-kramers relations in nuclear magnetic resonance

Abstract Experimental confirmation is presented for the applicability of the Kronig-Kramers relations (apart from a multiplicative scale factor) to saturated nuclear magnetic resonance systems obeying Redfield's relations. The materials chosen are metallic copper and aluminium, having particle sizes smaller than and greater than the skin depth. Further experimental evidence for the applicability of the modified Kronig-Kramers relations in non-metallic solids is also presented (27Al in beryl, 9Be in beryllium oxide).

Generalized immittance kernels and the Kronig-Kramers relations

A general relation between the poles and residues of a Kronig-Kramers pair is established which enables one to prove that the generalized immittance kernel obtained as the solution of an Nth-order linear differential equation with constant coefficients and sinusoidal forcing must necessarily be consistent with the Kronig-Kramers relations. Finally, it is proved that the network or system function of a system exhibiting a distribution over any or all of the (N + 1) arbitrary parameters of the Nth-order immittance kernel must itself be consistent with the Kronig-Kramers relations. The example of Lorentz dispersion is discussed.

Dielectric Dispersion in Materials Having a Distribution of Relaxation Times

On the basis of considerable experimental evidence, a particular form for the dielectric loss factor, ε″, is assumed to hold for many materials exhibiting a distribution of relaxation times. The dielectric constant, ε′, corresponding to the assumed form of ε″ is then computed by means of the Kronig-Kramers relations and tabulated as a function of log frequency and of the distribution function half-width parameter, α. Using these results, experimental ε′ and ε″ curves can be easily tested for mutual consistency and can be fitted individually. In conclusion, a brief discussion of other methods of dealing with the many-relaxation-time dispersion problem is given.

Line Shapes in Nuclear Paramagnetism

This note discusses briefly several methods of describing the width and shape of the resonances characteristic of nuclear paramagnetism. Use of the so-called moments of the shape function is illustrated by an experimental determination of a root mean square line width which is in substantial agreement with a calculation by Van Vleck. The solutions of the Bloch equations, which lead to the Lorentz shape function traditional in the theory of radiation- and collision-broadened lines, are compared with susceptibility curves based on a Gauss absorption curve. A significant difference between the two dispersion curves serves as a criterion for determining whether or not either of these two shapes closely approximates to a given experimental curve, and experimental examples of each are given.For comparison of shape functions, several well-known specializations of the Kronig-Kramers relations are employed. The collection of these formulae may incidentally prove convenient and useful to those engaged in studies of nuclear paramagnetism.