Cole-Davidson Research Articles

Evaluation of the Aging of Oil‐Paper Insulation Using an Improved Fractional Poynting‐Thomson Model

Assessment of the aging of oil‐paper insulation is analyzed in this article, based on an improved fractional Poynting‐Thomson model, which includes dc and hopping conductivity (P‐T model2). Based on samples of oil paper produced in laboratories with different thermal aging times, it has been found that the proposed improved model provides a better fit and a higher index of coincidence concerning the measured values than the traditional P‐T model and the model that only incorporates the effect of dc conductivity (P‐T model1). Nevertheless, comparing the proposed improved P‐T model with the traditional Cole–Cole (C‐C), Davidson–Cole (D‐C), and Havriliak–Negami (H‐N) models, the improved P‐T model2 also demonstrated a superior fit for the measured value and a higher coincidence index than another model. In addition, there was an excellent exponential relationship between the quantities εa, εb and τ of the improved model and the furfural content of the samples. Consequently, these characteristic quantities could be utilized to estimate the furfural content in the oil‐paper insulation and evaluate the insulation's condition. © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC.

Dynamically asymmetric binary glass formers studied by dielectric and NMR spectroscopy.

We investigate the component dynamics in asymmetric binary glass formers. Focusing on the dielectric spectra of the high-Tg components m-tricresyl phosphate and quinaldine mixed with toluene as low-Tg component, the broadend spectra cannot be described by Kohlrausch or Cole-Davidson (CD) functions. Instead, we apply a generalized CD function which allows to control the width of the susceptibility independently of its high-frequency flank. The spectra show a common broadening and failure of the frequency-temperature superposition with increasing toluene concentration. This is confirmed by stimulated echo experiments showing an increased stretching of the probed orientational correlation function. In analogy to the definition of Tg, we consider "isodynamic points". For each component, a different but linear concentration dependence of 1/Tiso is revealed, indicating different time scales. Qualitativly, we do not find significant differences for the present mixtures with Tg-contrasts of 63-89K compared to those with larger Tg-contrast ( [Formula: see text] K): Whereas the high-Tg component shows relaxation features similar to those of neat glass formers, yet, with "atypical" weak relaxation broadening, the faster low-Tg component displays pronounced dynamic heterogeneities. This is supported by scrutinizing NMR relaxation data of several mixtures investigated previously as a function of concentration. A universal evolution of the dynamics of the high-Tg as well as the low-Tg component is suggested for mixtures with high [Formula: see text]Tg .

Modeling of dielectric dissipation factor measurement for XLPE cable based on Davidson-Cole model

Dielectric response analysis is one approach for the condition assessment of high voltage insulation systems, and measurement of the dissipation factor or tan δ is widely used in practice. This article presents an experimental study of DF measurement over the frequency range from 1 mHz to 50 Hz. Three different short sections of service-aged 11-kV cross-linked polyethylene cable are used as test specimens. Low-voltage polarisation / depolarisation current and frequency domain spectroscopy techniques are employed in the investigation. Following the experimental work, numerical analysis is carried out utilizing the Davidson-Cole (D-C) model. To optimize the final parameters of the model, numerical simulation is performed using 3 or 4 R-C branches with different values of Jonscher and D-C constants. It is shown that results from numerical simulation agree well with the experimental results over the tested frequency range.

Modeling of dielectric dissipation factor measurement for XLPE cable based on Davidson-Cole model

Dielectric response analysis is one approach for the condition assessment of high voltage insulation systems, and measurement of the dissipation factor or tan δ is widely used in practice. This article presents an experimental study of DF measurement over the frequency range from 1 mHz to 50 Hz. Three different short sections of service-aged 11-kV cross-linked polyethylene cable are used as test specimens. Low-voltage polarisation / depolarisation current and frequency domain spectroscopy techniques are employed in the investigation. Following the experimental work, numerical analysis is carried out utilizing the Davidson-Cole (D-C) model. To optimize the final parameters of the model, numerical simulation is performed using 3 or 4 R-C branches with different values of Jonscher and D-C constants. It is shown that results from numerical simulation agree well with the experimental results over the tested frequency range.

Measurement of Cole–Davidson Diffusion Through Padé Approximations for (Bio)Impedance Spectroscopy

Throughout various engineering applications, measurement science faces the problem that parameters of interest are not directly measurable with a specifically designed instrument. As such, one relies on indirect measurement which is mapped through a mathematical model to the parameters of interest. The parameters of interest are consequently estimated statistically through analyzing the measurements. This field is known as system identification and serves applications in control, mechanical, and biomedical engineering among others. Anomalous diffusion pops up in impedance and dielectric spectroscopy such that the application of system identification techniques in electrochemical, microwave, and biomedical engineering may be affected by diffusion. The presence of diffusion may introduce systematic errors or bias when applying system identification techniques of linear time-invariant (LTI) systems. In this paper, we revisit classical LTI identification in the presence of Cole-Davidson (CD) diffusion wherein we accomplish: 1) detecting the CD diffusion component; 2) discriminating the diffusion from the remaining dynamics; and 3) modeling the CD diffusion through a fractional-order model, in particular, a pole of fractional-order multiplicity.

A discussion on robust multi-variable I/O fractional transfer function of types DC or FBLFD in motion control

Motion control and robust path tracking design are the objective of this paper. The Multi-Input Multi-Output Quantitative Feedback Theory (MIMO-QFT) method is considered. This methodology permits to divide the MIMO system into sub-Multi-Input-Single-Output (MISO) equivalent structures. After obtaining the sub-structures, the objective is the design of the controller and prefilter in order to achieve some specifications. A multi-variable robust path tracking design is already studied based on fractional approaches. These developed approaches are based on two fractional prefilters of types Davidson Cole (DC) and Frequency Band Limited Fractional Differentiator (FBLFD). A new methodology that combines I/O (Input Output) fractional transfer function of types DC and FBLFD with a fractional PID controller tuning is studied in this paper. This approach is based on considering the I/O transfer function matrix as a fractional prefilter of types DC or FBLFD. Using this approach a reduced settling time of the I/O transfer function can be obtained. The multivariable fractional PID controller is tuned based on a multi-objective optimisation using genetic algorithm. The obtained method is tested with a SCARA robot model to demonstrate the efficacy and robustness of our approach.

Consideration of dielectric relaxation of pure DMSO liquid in different temperatures

This paper mainly analyzes the relaxation process of both CC model and Davidson–Cole (DC) model, depicting the process by relaxation time, broadening parameter and temperature dependent Kirkwood correlation factor. The Kirkwood correlation factor of CC model in adjustable infinite dielectric constant is more changeable with temperature than DC model. The Kirkwood factor with different conditions needs reinterpretation. CC model generally depicts molecular cooperative interactions with single channel and evaluates once relaxation act of DMSO cluster within a cutoff time. DC model couples multi-channels, including various relaxation modes such as monomeric, dimeric and cluster. So the relaxation time of DC model is higher than CC model.

Comparison between Davidson-Cole and Frequency-Band Limited Fractional Differentiator I/O Type Transfer Function with Speed and Acceleration Inputs in Path Tracking Design

A new approach to path tracking design based on fractional prefilter was developed in this paper. In path tracking design, the dynamic of actuators must be taken into account in order to reduce over- shoots appearing for small displacements. Taking into consideration the maximum velocity, acceleration, jerk, and the bandwidth of the closed-loop on which the input is applied, it permits the generation of an optimal movement reference-input giving a minimum path comple- tion time. An approach to path tracking based on fractional prefilter has been developed. This approach based on a Davidson-Cole (DC) and Frequency Band Limited Fractional Differentiator (FBLFD) pre- filters, with position input. This work describes an extension of this method. It consists of a path tracking using fractional differentiation and comparison between different types of prefilters by direct opti- mization of an Input/Output (I/O) transfer function with speed and acceleration inputs. Fractional differentiation has been used through a Davidson-Cole and frequency band-limited fractional differentiator (FBLFD) prefilters. A simulation on a motor model validates the developed methodology.

Electron transfer in a two-level system within a Cole-Davidson vitreous bath

We study electron transfer (ET) in a two level quantum system coupled to a glassy viscous bath. The bath is modeled by the Cole-Davidson (CD) spectral density. The ET in this model is compared to the ET in a normal Drude-Debye (DD) model. It is shown that at low temperatures and when the coupling to the bath is weak, the viscous bath preserves the quantum coherence for a longer time. However in the strong coupling regime, the tunneling rate is higher in the CD. In the classical high temperature limit the difference between the CD and DD models is negligible.

The impact of hierarchically constrained dynamics with a finite mean of cluster sizes on relaxation properties

In this paper, a stochastic scenario of relaxation underlying the generalization (Kahlau et al., 2010) [15] of the Cole–Davidson (CD) and Kohlrausch–Williams–Watts (KWW) functions is proposed. As it has been shown (Kahlau et al., 2010) [15], the new three-parameter time-domain fitting function provides a very flexible description of the dielectric spectroscopy data for viscous glass-forming liquids. In relation to that result we discuss a hierarchically-constrained model yielding the proposed relaxation fitting function. Within the “exponentially decaying relaxation contributions” framework we show origins of the high-frequency (short-time, respectively) fractional power law, i.e., the characteristic feature of the new, as well as, of both CD and KWW response functions. We also bring into light a reason for which their common behavior in the opposite frequency limit is linear on external field frequency. Finally, we relate the new relaxation pattern (Kahlau et al., 2010) [15] with the Generalized Gamma (GG) survival probability of an imposed, non-equilibrium initial state in a relaxing system.

Study of dielectric relaxation and hydrogen bonding in water + 2-butoxyethanol mixtures using TDR technique

The complex permittivity of 2-butoxyethanol (BE)–water mixtures over entire concentrations has been measured as a function of frequency from 10MHz to 30GHz. All spectra were fitted using Cole–Davidson (CD) relaxation spectral function which gives an asymmetric distribution of relaxation times. As composition of butoxyethanol in water increases the width of distribution function broadens. By using least squares fit method the dielectric parameters such as static dielectric constant (ɛ0), dielectric constant at high frequency (ɛ∞), relaxation time (τ) and relaxation distribution parameter (β) were extracted from complex permittivity spectra at temperature range from 25°C to 0°C. The heterogeneous interaction in unlike molecules and intramolecular interaction in same molecules has been discussed using the excess dielectric properties, Kirkwood correlation factor, thermodynamic properties and Bruggeman factor.

Generalization of the Cole–Davidson and Kohlrausch functions to describe the primaryresponse of glass-forming systems

We introduce a three-parameter step-response function which is based on a generalizationof the Cole–Davidson (CD) and Kohlrausch (K) functions, and which provides ahighly flexible susceptibility description for viscous liquids. A second parameterα characterizing the overall width, in addition to a parameterβ determining the high-frequency behavior of the susceptibility, allows for acontinuous change of the spectral shape from the CD-type to the K-type.We prove that the function fulfills mathematical conditions required for astep-response function. When applying the function to interpolate dielectric spectra ofneat (pure) glass formers, it is possible to keep the high-frequency parameterβ temperature-independent while varying the parameterα to account for the change of the overall width. This analysis might suggest that thefailure of frequency–temperature superposition in glass formers is reflected bya broadening in the low-frequency region instead of in the high-frequency one.

Comparative Studies of the Dynamics in Viscous Liquids by Means of Dielectric Spectroscopy and Field Cycling NMR

(1)H nuclear magnetic resonance (NMR) field cycling relaxometry has been applied to study the dynamics of homologues of glycerol (propylene glycol, 2,3-butandiol, glycerol, threitol, xylitol, sorbitol) and non-alcohol liquids (o-terphenyl, tristyrene, 3-fluoroaniline, m-toluidine). A broad range of temperatures (200-400 K) is covered, including the supercooled state. From the dispersion of the spin-lattice relaxation rate, R(1)(omega) = T(1)(-1)(omega), the susceptibility representation, omegaT(1)(-1)(omega), has been derived and compared with the susceptibility data from dielectric spectroscopy (DS). The DS spectra can be interpolated by a Cole-Davidson (CD) function, yielding correlation times which are attributed to the structural relaxation (alpha-process) in the supercooled state. In contrast to that, most of the (1)H NMR susceptibility spectra show, in addition to the alpha-relaxation peak, a low-frequency excess contribution with amplitudes varying among the systems. Exceptions are o-terphenyl and tristyrene for which DS and NMR susceptibility curves agree well and both can be reproduced by a CD function. Possible explanations of the low-frequency contribution are discussed. In particular the role of translational diffusion probed via the intermolecular coupling of (1)H spins is considered since it may likely generate the low-frequency excess intensity.

Improved Dielectric Model for Polyvinyl Alcohol-Water Hydrogel at Microwave Frequencies

Problem statement: The study described rigorous dielectric modeling for Polyvinyl Alcohol (PVA)-water hydrogel mixture at microwave frequencies. Approach: A commercial open-ended coaxial sensor was used to measure the dielectric constant, loss factor and ionic conductivity, σ of PVA-water hydrogel mixture ranging concentration of 80-100% water content. Results: The sensor was operating between 0.13 and 20 GHz at and above of room temperature (25±1)°C. Indirectly, the relaxation time, τ, activation energy, Q and entropy change, ΔS of the hydrogel mixtures are determined based on linear fitting of measured data using Debye and Arrhenius approaches. Conclusion/Recommendations: Two main relaxation processes were found ranging 2-10 and 10-20 GHz, respectively. Dielectric dispersion is suggested to describe by combination of Cole-Davidson (CD) and Debye (Dy) processes. The results are discussed qualitatively based on bound states of water in hydrogel mixtures.

Molecular Dynamics and Entropy Effects in Hydrogen-Bonded Supramolecular PolymerN,N′-Di(2-methyl-2-pentylheptyl)urea Dissolved in Nonpolar Medium

On the basis of the static dielectric permittivity temperature behavior of the supramolecular polymer formed by N,N'-di(2-methyl-2-pentylheptyl)urea (MPHU) in carbon tetrachloride diluted solution (10% in mole fraction) and that of neat acetonitrile, it was found that two liquids of the same macroscopic polarity (expressed by the same value of the dielectric permittivity approximately 35) exhibit an essential difference in the field-induced orientational entropy change. A much higher entropy effect is observed for the liquid composed of not too numerous but highly polar molecular entities (MPHU + CCl(4)) than for the liquid composed of numerous but less polar entities (neat acetonitrile). The analysis of the dielectric relaxation spectra of MPHU + CCl(4) solutions was performed with the Davidson-Cole (DC) model. It was found that the DC exponent beta changes its value in a quite important range of 0.4-0.8, depending on MPHU concentration and temperature. This reflects an important deviation of the relaxation mechanism occurring in the supramolecular system investigated from that corresponding to the normal Brownian rotational diffusion (beta = 1).

The justified data-curve fitting approach: recognition of the new type of kinetic equations in fractional derivatives from analysis of raw dielectric data

Usually, for the description of dielectric spectra one uses the empirical Cole–Davidson (CD) and Havriliak–Negami (HN) equations each of which contains one relaxation time. However, the parameters figuring in the CD and HN equations (or the linear combination of several CD or HN equations) do not have any clear physical meaning. For the description of such asymmetric dielectric spectra, we suggest complex permittivity functions containing two or more characteristic relaxation times. These complex susceptibility functions correspond, in the time-domain, to a new type of kinetic equation, which contains non-integer (fractional) integrals and derivatives. The physical meaning of these operators is discussed in [1]. We suppose that these kinetic equations describe a wide class of dielectric relaxation phenomena taking place in heterogeneous substances. To support and justify this statement, a special recognition procedure has been developed that helps to identify this new kinetic equation from real dielectric data. This recognition procedure can be considered as the justified data-curve fitting (JDCF) approach, in contrast to the conventional ‘imposed’ data-curve fitting (IDCF) treatment invariably used in modern dielectric spectroscopy. The JDCF approach incorporates the ratio presentation (RP) format and a separation procedure. It is shown how this separation procedure can be helpful in the detection of the many relaxation processes (each process is described by a characteristic relaxation time), which are taking place in the dielectric material under consideration.

Signal processing and recognition of true kinetic equations containing non-integer derivatives from raw dielectric data

Signal processing in dielectric spectroscopy implies that it is necessary to find a ‘true’ fitting function (having a certain physical meaning), which describes well the complex permittivity and impedance data. In dielectric spectroscopy for description of complex permittivity/impedance data researches usually use the empirical Cole–Davidson (CD) and Havriliak–Negami (HN) equations that contains one relaxation time. But the parameters figuring in CD and HN equations do not have clear physical meaning as well as fitting parameters entering into linear combination of several CD or HN equations. For description of dielectric (especially asymmetric) spectra we suggest the complex permittivity functions containing two or more characteristic relaxation times. These complex susceptibility functions correspond in time domain to new type of kinetic equation containing non-integer (fractional) integrals and derivatives. We suppose that these kinetic equations describe a wide class of dielectric relaxation phenomena taking place in heterogeneous substances. To support and justify this statement the special recognition procedure has been developed that helps to identify this new kinetic equation from raw dielectric data. It incorporates the ratio presentation (or RP) format and separation procedure. Separation procedure was turned out to be helpful in detection of number of relaxation processes (each process is described by a characteristic relaxation time) taking place in the dielectric material under consideration. We suppose that this procedure can be applicable also for identification of fractal noises.

Physical mechanisms responsible for the relaxation time distribution in disordered dielectrics

The relaxation time distribution function F(τ) is calculated in the framework of the random-field theory. The function F(τ) is expressed through the distribution function f(E) of a random electric field E with due regard for the derived dependence of the relaxation time τ on the electric field. The distribution function F(τ) is calculated in terms of the statistical theory within the random-field approximation. The nonlinear random-field contributions and spatial effects of correlations between randomly distributed electric dipoles are taken into account. The calculations are performed for a mixed ferroelectric glassy phase in which the short-range and long-range polar orders coexist. It is demonstrated that the inclusion of nonlinear contributions of the random field leads to an asymmetric relaxation time distribution function F(τ), whereas allowance made only for the linear random-field contributions results in a symmetric function F(τ). A comparison of the calculated functions F(τ) with empirical functions derived from the Cole-Cole (CC), Davidson-Cole (DC), Kohlrausch-Williams-Watts (KWW), and Havriliak-Negami (HN) laws for the dielectric response shows that these laws correspond to disordered systems in which the long-range and short-range orders coexist. Different forms of the function F(τ) are determined by either linear (the CC law) or nonlinear (the DC, KWW, and HN laws) contributions of the random field.

Relaxation time distribution function

The distribution function of relaxation time in disordered dielectrics has been calculated in the random field theory framework. For this purpose, we first consider the dynamics of single two-orientable electric dipole in a random electric field E in a disordered ferroelectric. The obtained dependence τ(E) made it possible to express the relaxation time distribution function F(τ) via the random field distribution function f(E). This function has been calculated self-consistently in the random field theory framework. Nonlinear random field contribution and effects of spatial correlations between the dipoles have also been taken into account. It was shown that nonlinear contribution of random field gives asymmetric shape of F(τ), while in linear case it is symmetric. Comparison of calculated F(τ) curves with those extracted from empirical Cole-Cole (CC), Davidson-Cole (DC), Kohlrausch-William-Watts (KWW) and Havriliak-Negami (HN) functions had shown, that they correspond to mixed ferroelectric-dipole glass (ferro-glass) phase with coexistence of short and long-range order. Different forms of F(τ) are determined by linear (CC) or nonlinear (DC, KWW, HN) contributions of random field.

Random field based theory of the relaxor ferroelectrics

We present the random field based theory of relaxor ferroelectrics such as PMN, PSN and PLZT. We consider relaxors as systems of defects of different type (e.g. random site and orientation electric dipoles, lattice vacancies etc) embedded into Burns reference phase with long-range ferroelectric order; this order is being destroyed by the aforementioned defects. The equations for description of relaxors phase diagram as well as other observable properties like static or dynamic linear and nonlinear dielectric susceptibility has been derived. These equations also permit to obtain the relaxation times distribution function in relaxors. The latter function has been calculated for different parameters of relaxor ferroelectrics, corresponding to several well-known empirical relaxation laws like Cole-Cole (CC), Davidson-Cole (DC) and Kohlrausch-Williams-Watts (KWW). Fairly good agreement between theory and experiment has been achieved.