Distributions from Time-Domain Measurements

3 C Distance Distributions from Time-Domain Measurements 

The uppo panel in Hgme 14.16 shows the time-dependent don decays of the doni -alone protein and the donor-acceptor protein in tiie absence of bound calcium. Witiiout acc tor, tiie intensky dec of trp-22 is close to a single ei nential. Die intensity decay of trp-22 be- 

Flgpire 14.16. Tiyptophan-22 intensiQr decays of troponin C mutant (F22W/N52OCI0IL) without (D) and with (DA) 1ABE ANS on cysteine-52. Top Without Ca Bottom with Ca. Prom Ref. 9. 

It is informative to examine the multiexponential analyses of the donor decays (Table 14.2). In the absence of acceptor, the tryptophan decays are close to single exponentials, especially in the atsence Ca. hi the pcesmce of the lA ANS acc tor, the intensity decay becomes strongly heterogeneous. The extent of het geneity can be jud the xi values for the single-decay-time fits. 

The wider distribution in the absence of Ca results in a hi er wilue of (xi = 192) tiian in the presen of (Xff = 32), where the distribution is narrower. The intensity decays in the absence and presence of reveal compo- 

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D. Analysis of Distance Distribution from Time-Domain Measurements... 

The two quantities which can be observed when an individual positron annihilates in condensed matter are the positron age r, which is the time interval between implantation and annihilation of the positron, and the momentum p of the annihilating positron-electron pair. Time-resolved information on the evolution of positron states is obtained by correlated measurements of the individual positron lifetime (= positron age) and the momentum of the annihilating positron-electron pair (Age-Momentum Correlation, AMOC). AMOC measurements are an extremely powerful tool for the study of reactions involving positrons. It not only provides the information obtainable from the two constituent measurements but allows us to follow directly, in the time domain, changes in the e+e momentum distribution of a positron state (cf. Sect. 1). 

Fig. 15b shows the value of the integral and the amplitude 1 - Ctp as function of im, where the latter was determined from a suitable fit of the correlation functions. The good agreement indicates that the amount of correlation lost in the experimental time window provides a measure of the location of Gflg t). We emphasize that these results are not limited to NMR correlation functions, but they show that any time-domain experiment will not provide straightforward access to the mean time constant of a dynamical process that is governed by a distribution G lg t) broader than the experimental time window. 

The extent of gas dispersion can usually be computed from experimentally measured gas residence time distribution. The dual probe detection method followed by least square regression of data in the time domain is effective in eliminating error introduced from the usual pulse technique which could not produce an ideal Delta function input (Wu, 1988). By this method, tracer is injected at a point in the fast bed, and tracer concentration is monitored downstream of the injection point by two sampling probes spaced a given distance apart, which are connected to two individual thermal conductivity cells. The response signal produced by the first probe is taken as the input to the second probe. The difference between the concentration-versus-time curves is used to describe gas mixing. 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

When the fiuctuations show current or voltage transients, data analysis may be performed in the time domain by investigating the shape, size and occurrence rate of the random events. It can be also performed by measuring the moments of the potential or current fiuctuations (standard deviation, skewness, kurtosis) [12]. However, this approach is extremely limited for data interpretation. In the absence of current or voltage transients, the values of the moments are most likely close to zero, as for signals with a Gaussian distribution. Any deviation from zero indicates the existence of transients. 

The primary result of a pulsed ELDOR measurement is a distribution of dipolar cou-phngs d. This information is contained either in the primary time-domain data (variation of echo intensity as a function of dipolar evolution time t) or in the dipolar spectrum obtained from these data by Fourier transformation. Time- and frequency-domain data contain exactly the same information, since Fourier transformation is a linear operation. However, some features are easier to recognize in time domain (e.g., quahty of least-squares fitting of the data) and others in frequency domain (e.g., orientation selection by missing parts of the Pake pattern). 

ABSTRACT State determination of Li-ion cells is often accomplished with Electrochemical Impedance Spectroscopy (EIS). The measurement results are in frequency domain and used to describe the state of a Li-ion cell by parameterizing impedance-based models. Since EIS is a costly measurement method, an alternative method for the parameterization of impedance-based models with time-domain data easier to record is presented in this work. For this purpose the model equations from the impedance-based models are transformed from frequency domain into time domain. As an excitation signal a current step is applied. The resulting voltage step responses are the model equations in time domain. They are presented for lumped and derived for distributed electrical circuit elements, i.e. Warburg impedance, Constant Phase Element and RCPE. A resulting technique is the determination of the inner resistance from an impedance spectrum which is performed on measurement data. 

Due to the high measurement and computational complexity as well as cost factors, frequency domain EIS measurements are not likely to be implemented on board in vehicles in the near future. An alternative approach is shown to parameterize impedance-based models with time domain data available on board, i.e. currents, battery or cell voltages and temperatures. Therefore, in this work, a method is proposed for the transformation of electrical circuit model equations from frequency domain into time domain model equations. Particularly for electrical circuit models containing distributed elements, e.g. Warburg impedances (WB), Constant Phase Elements (CPE), RCPE or ZARC elements, these transformations require fractional calculus methods, as will be presented in detail. 

A method was proposed for the parameterization of impedance based models in the time domain, by deriving the corresponding time domain model equation with inverse Laplace transform of the frequency domain model equation assuming a current step excitation. This excitation signal has been chosen, since it can be easily applied to a Li-ion cell in an experiment, allows the analytical calculation of the time domain model equation and is included in the definition of the inner resistance. The voltage step responses of model elements were presented for lumped elements and derived for distributed model elements that have underlying fractional differential equations using fractional calculus. The determination of the inner resistance from an impedance spectrum was proposed as a possible application for this method. Tests on measurement data showed that this method works well for temperatures around room temperature and current excitation amplitudes up to 10 C. This technique can be used for comparisons of measured impedance spectra with conventionally determined inner resistances. 

The distributed signals can be measured and spatially resolved by an optical time-domain reflectometry technique. An optical time domain reflectometer is based on the measurement of backscattered light attained from a light pulse propagating through an optical fiber. Light is backscattered because of inhomogeneities and impurities... 

Via a passive scalar method [6] where or, denotes the volume fraction of the i-th phase, while T, represents the diffusivity coefiBcient of the tracer in the i-th phase. The transient form of the scalar transport equation was utilized to track the pulse of tracer through the computational domain. The exit age distribution was evaluated from the normalized concentration curve obtained via measurements at the reactor outlet at 1 second intervals. This was subsequently used to determine the mean residence time, tm and Peclet number, Pe [7]. 

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