Davidson-Cole Model Research Articles

A New Look at the Capacitor Theory

The mathematical description of the charging process of time-varying capacitors is reviewed and a new formulation is proposed. For it, suitable fractional derivatives are described. The case of fractional capacitors that follow the Curie–von Schweidler law is considered. Through suitable substitutions, a similar scheme for fractional inductors is obtained. Formulae for voltage/current input/output are presented. Backward coherence with classic results is established and generalised to the variable order case. The concept of a tempered fractor is introduced and related to the Davidson–Cole model.

Assessment on Oil-Paper Insulation Aging of Transformer Based on Dielectric Response Model

For nondestructive assessment of insulation aging in power transformer, dielectric response technology is usually used to evaluate the oil-paper insulation aging. In this paper, a quantitative evaluation method of insulation aging based on Davidson-Cole model is proposed. The oil-paper insulation equivalent Debye model parameters of transformer are extracted by exponential function fitting. According to the complex permittivity of oil-paper insulation, the parameters of Davidson-Cole model are extracted by fitting algorithm. Under laboratory conditions, accelerated thermal aging test are carried out. The quantitative relationship between Davidson-Cole model parameters and aging of oil-paper insulation is analyzed. The experimental results show that the Davidson-Cole model parameters are directly related to the aging time. It is found that the parameters β and Δε of the Davidson-Cole model gradually decreases and the parameter τ gradually increases with the polymerization degree of oil-impregnated paper. Therefore, according to the quantitative characterization of the model parameters, the aging of transformer oil-paper insulation can be effectively evaluated, which is helpful to realize the nondestructive testing of polymerization degree in oil-impregnated pressboard (paper) power transformer.

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.

Frequency electrical conductivity dependence and dielectric relaxation of ZnO–P2O5/Co composites

This work focuses on the experimental study of the conductivity and the permittivity of ZnO–P2O5/Co (ZP/Co) composites as function of frequency in the range of 100–106 Hz at room temperature. The conductivity evolution shows a high percolating jump from low to high loaded ZP matrix with Co filler, indicating an insulator–semiconductor phase transition. Above the percolation threshold, the obtained conductivity is between 10−1 and 10° S/cm and is frequency independent. While, for low filler volume fraction, it is frequency dependent and it is about 10−8 S/cm at 100 Hz. Moreover, it should be noticed that around the percolation threshold, a significant value of dielectric constant about 105 was observed at room temperature and low frequency 100 Hz. Furthermore, the distribution of the relaxation processes was also investigated below the percolation threshold. Indeed, a relaxation peak about 106 rad/s in the curves of electric modulus M″(ω) was observed. It is successively described by Davidson–Cole model and an account of the interpretation of obtained results is given.

A note on applications of time-domain solution of Cole permittivity models

The complex dielectric permittivity is an important parameter for characterization of electrical properties of dielectric materials. Experimental studies demonstrated that Cole models of dielectric permittivity can provide a better fitting to the experimental data because of allowing for fractional-order frequency dependence. This study aims to investigate physical interpretation of time domain solutions of Cole permittivity models. For this purpose, impulse responses of Cole-Cole model and Davidson-Cole model are expressed in Mittag-Leffler function form by using inverse Laplace transform. The impulse responses of these models are decomposed into impulsive and dispersive components, and the relations of these components with relaxation mechanism of dielectric materials are discussed. By considering impulse response solution of Cole-Cole models, a fractional order dynamic capacitance model is introduced for time domain equivalent circuit modeling of dielectric materials. Moreover, transient properties of electromagnetic wave penetration to dielectric materials are analyzed according to impulse response solution of Cole-Cole model. To illustrate applications of proposed time domain permittivity solutions, the Cole-Cole model of ethyl-acetate liquids was also studied and results are presented.

Dielectric properties of 1-methylimidazole at temperatures 298–336 K

ABSTRACT Dielectric permittivity and loss spectra of 1-methylimidazole (MeIm) have been measured over the frequency range from 100 MHz to 20 GHz at 298–336 K using an open-ended coaxial probe technique. Dielectric parameters are described by the Davidson–Cole model. The shape parameter shows temperature-independence. The Kirkwood correlation factor (g), enthalpy, entropy and the activation energy of molecular reorientation of MeIm also have been calculated.

Ac-conductivity and dielectric response of new zinc-phosphate glass/metal composites

The ac-conductivity and dielectric response of new composites based on zinc-phosphate glass with composition 45mol%ZnO–55mol%P2O5, filled with metallic powder of nickel (ZP/Ni) were investigated by impedance spectroscopy in the frequency range from 100Hz to 1MHz at room temperature. A high percolating jump of seven times has been observed in the conductivity behavior from low volume fraction of filler to the higher fractions, indicating an insulator – semiconductor phase transition. The measured conductivity at higher filler volume fraction is about 10−1S/cm and is frequency independent, while, the obtained conductivity for low filler volume fraction is around 10−8S/cm and is frequency dependent. Moreover, the elaborated composites are characterized by high dielectric constants in the range of 105 for conductive composites at low frequencies (100Hz). In addition, the distribution of the relaxation processes was also evaluated. The Debye, Cole-Cole, Davidson–Cole and Havriliak–Negami models in electric modulus formalism were used to model the observed relaxation phenomena in ZP/Ni composites. The observed relaxation phenomena are fairly simulated by Davidson–Cole model, and an account of the interpretation of results is given.

Hydrogen bond analysis in alcohol (1-Propanol, 2-Propanol and Glycerol)-DMF mixtures based on dielectric spectroscopy

Dielectric properties of Propanol-DMF, and Glycerol-DMF mixtures at full concentrations have been obtained by the dielectric relaxation spectroscopy method at frequency from 20 MHz to 20 GHz at room temperature. The mixture behavior is described by the Davidson-Cole model using four parameters that relate to molecular structure. The hydrogen bond (HB) numbers and binding energies between two pairs (solute–solute and solute-solvent pairs) are estimated by Luzar model, also the type of dipolar ordering which depends on the extent of DMF is obtained. The combination of the excess inverse permittivity and the Luzar model gives a satisfactory explanation and valuable insights into the underlying of the relaxation acts. The binding energy of solute–solute (E11) and solute-solvent (E12) both decrease with the increased number of hydroxyl in mixture system.

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.

The hydrogen bonding dynamics and cooperative interactions of NMP–water mixture studied by dielectric relaxation spectroscopy

The permittivity for N-Methyl-Pyrrolidone (NMP)–water mixtures has been determined from frequency 20MHz to 26GHz at 298K using dielectric relaxation spectroscopy (DRS). The dielectric behavior is described well by Davidson–Cole model. The excess inverse relaxation time and excess inverse permittivity describe a hydrogen bond effect for molecular relaxation time and interactions. The Luzar model provides an explanation of experimental results related to the static permittivity and estimates numbers of hydrogen bond in the mixture. We parameterize NMP–water cluster interactions providing insights into the underlying mechanism of relaxations.

Rietveld refinement and dielectric relaxation of a new rare earth based double perovskite oxide: BaPrCoNbO6

A new rare earth based double perovskite oxide barium praseodymium cobalt niobate, BaPrCoNbO6 (BPCN) is synthesized by solid-state reaction technique. Rietveld analysis of X-ray diffraction (XRD) data shows that the compound crystallizes in a perovskite like tetragonal structure which belongs to the I4/mmm space group with lattice parameters a=b=5.6828(9)Å, c=8.063(2)Å. Structural analysis reveals 1:1 ordered arrangement for the Co2+ and Nb5+ cations over the six-coordinate B-sites of BPCN. The superlattice line (101) at 2θ=19.10° arising from the alternate ordering of Co2+ and Nb5+ sites is observed in the XRD pattern which confirms the presence of cation ordering in BPCN. Fourier transform infrared spectrum shows two phonon modes of the sample due to the antisymmetric NbO6 stretching vibration. The relaxation dynamics of the conductive process in BPCN is investigated in the temperature range 303 to 503K and in the frequency range 100Hz to 1MHz using impedance spectroscopy. The relaxation mechanism of the sample in the framework of electric modulus formalism is modeled by Davidson–Cole model (DCM). The values of α (distribution of relaxation time) for the DCM varies from 0.1 to 0.3 which suggests the asymmetric distribution of relaxation time for BPCN. The activation energy of the sample, calculated from both conductivity and modulus spectra, are found to be almost the same ~0.4eV, which indicates that the conduction mechanism for BPCN is polaron hopping. The scaling behaviour of the imaginary part of electric modulus suggests that the relaxation follows the same mechanism at various temperatures.

Dielectric Relaxation of Curing DGEBA/mPDA System at 2.45 GHz

The dielectric behavior of a curing system of diglycidyl ether of bisphenol A and m-phenylenediamine has been studied over the temperature range of 20—100°C under 2.45 GHz microwave radiation. The dielectric constant and the dielectric loss factor of the system increase as temperature increases while they decrease as the curing reaction progresses. The epoxy resins at different extents of cure exhibit the g relaxation, which can be described by the Arrhenius rate law. The relaxation is attributed to the motions of the dipolar groups associated with the crosslinking system. The Davidson—Cole model is used to represent the temperature dependency of the dielectric properties.

Dielectric studies of three epoxy resin systems during microwave cure

Dielectric properties of three curing epoxy resin systems at an industrial microwave frequency, 2.45 GHz, were measured over a temperature range lower than the cure temperature. Extent of cure, which is determined by DSC, is used to describe the progress of the polymerization. It has been found that, normally, the real and the imaginary part of the complex dielectric constant increased with temperature and decreased during microwave cure. The changes of the dielectric properties during the reaction are related to the decreasing number of the dipolar groups in the reactants and the increasing viscosity. The Davidson–Cole model can be used to describe the experimental data. The Zong model is applicable to polymeric materials at high microwave frequencies and can be used to calculate the parameters of the Davidson–Cole model. The epoxy resins exhibit one γ relaxation, which can be described by the Arrhenius rate law. The evolutions of the parameters in the models are discussed.

Dielectric properties of an epoxy-amine system at a high microwave frequency

The dielectric properties of a curing diglycidyl ether of bisphenol A (DGEBA)/Jeffamine D-230 system have been studied over the temperature range of 20–90°C at 2.45 GHz. It was found that, generally, both the dielectric constant and the dielectric loss factor of the system increased with temperature and decreased as the reaction proceeded. The epoxy resins at different extents of cure exhibited the γ relaxation, which can be described by the Arrhenius Rate Law. The relaxation is attributed to the motions of the dipolar groups associated with the reactants. The Davidson-Cole model can represent the temperature dependencies of the dielectric properties. The nature of the information yielded by dielectrometry on the dynamics of the system is discussed. The evolution of the parameters of the models during the polymerization was mainly affected by the decreasing number of the dipolar groups involved in the reaction and increasing medium viscosity. POLYM. ENG. SCI. 45:1576–1580, 2005. © 2005 Society of Plastics Engineers

A percolation cluster model of the temperature dependent dielectric properties of hydrated proteins

This study investigates the temperature dependence of the low frequency dielectric properties (0.1 Hz–1 MHz) of hydrated globular proteins (namely, ovalbumin, lysozyme and pepsin). The study aims to reveal the mechanisms of water–protein interaction from the dielectric response of these model proteins. Two principle dielectric responses were observed for each hydrated protein, namely, an anomalous low frequency dispersion and a dielectric loss peak at higher frequency (called the ε3 dispersion). The low frequency response conformed to a fractional power low of frequency, while the higher frequency response conformed to a Davidson–Cole model. The strength of both processes reached a maximum at a certain temperature within the experimental temperature range. This temperature is referred to as the percolation threshold (PT) and is thought to be associated with the percolation of protons between hydrogen-bonded water molecules. The relaxation times of the ε3 dispersion conformed to Arrhenius behaviour at temperatures below the PT, from which an activation energy (ΔH) could be calculated. This activation energy is thought to be a measure of the concentration of available charged sites through which proton transport is facilitated. The structural fractal dimension in the hydrated protein system was also calculated, and enabled the approximation of the pathway for charge percolation in the protein matrix.

1H and 13C NMR Studies of Molecular Dynamics in the Biocopolymer of Glycolide and ε-Caprolactone

Copolymers of glycolide and ε-caprolactone were studied using differential scanning calorimetry and solid-state NMR. The variation of the T 1 relaxation time with temperature reflects local disorder and can be quantified in terms of the distribution of correlation times predicted by the Davidson–Cole model. T 1 relaxation is dominated by trans–gauche isomerisation, with an activation energy of 34–35 kJ mol −1.

Molecular dynamics in supercooled m-fluoroaniline

Molecular dynamics of meta-fluoroaniline (mFA) — the system which makes a simple glass forming liquid — was examined by 13C NMR and dielectric relaxation measurements. Spin–lattice relaxation times T1, nuclear Overhauser (NOE) factors of chemically non-equivalent carbons and dielectric relaxation in mFA were measured as a function of temperature. The analysis showed that the correlation function describing molecular dynamics in mFA could be well described in terms of an asymmetric distribution of correlation times predicted by the Davidson–Cole model. The overall correlation times of mFA, derived from the simultaneous fitting of our T1, NOE, and dielectric data (as well as of the dielectric data obtained by Roessler's group) to the Davidson–Cole model, follow the non-Arrhenius temperature type of the behaviour in accordance with the Vogel–Fulcher equation.

Nuclearmagnetic relaxation study of carbon-13 in di-pentyl phthalate

Spin-lattice relaxation times T1 and nuclear Overhauser enhancement factors of chemically non-equivalent carbons in di-pentyl phthalate (DPP), a simple glass-forming liquid, are measured as functions of temperature. The analysis shows that the correlation function describing the overall molecular dynamics in DPP can be well described in terms of an asymmetric distribution of correlation times predicted by the Davidson-Cole model. The distribution parameters of the successive chain carbons are derived and analysed in terms of internal motions occurring in DPP.

Loss factor height and width limits for polymer relaxations

Polymer relaxations at the glass transition are often used in damping applications. Questions arise whether there is any limit to the height and width of the damping peak that can be achieved. A related question is whether there are any limits on the combination of these two properties that are achievable. This later question arises because of the experimental observation that the height of the peak is inversely related to the width of the peak. In this paper, these questions are addressed using various analytical models of polymer behavior. Starting with the single relaxation time model and progressing to the Cole–Cole model, the Davidson–Cole model, and finally the Havriliak–Negami (HN) model, height and width predictions are obtained. It is found that the HN model predicts a band of physically possible height and width combinations when using reasonable values of the model parameters. High peaks are narrow and broad peaks are low. For a loss factor peak height of 2, the half-width must be less than 3 decades; for a half-width of 8 decades, the peak height must be less than 0.5. A comparison with experimental data for polymers of widely varying properties is in good agreement with these predictions.