Harmonic analyses

Harmonic analysis, or Fourier analysis, is the decomposition of a periodic function into a sum of simple periodic components. In particular, Fourier series are expansions of periodic functions /(x) in terms of an infinite sum of sines and cosines of the form 

The coefficients a and b can be obtained by the orthogonal property of the sine and cosine functions 

Similarly, we obtain the coefficient a by multiplying by cos(nx) and integrating. The Fourier series of a linear combination of two functions is the linear combination of the corresponding two series. If /(x) periodic with period 2L, with an appropriate transformation again the formulas can be used. If the period goes to the limit L 00, then the Fourier integral emerges. 

Fourier did his important mathematical work on the theory of heat from 1804 to 1807. The reason why he was engaged on the theory of heat was purely personal, namely he suffered from rheumatic pains. Contemporaries report that he kept his dwelling so hot since his stay in Egypt that visitors could not bear the hotness, while he wrapped himself additionally still into thick coats. His memoir On the Propagation of Heat in Solid Bodies was at that time accepted with controversy, but is nowadays highly commended. In 1808, Lagrange and Laplace opposed the idea of trigonometric expansion. 

The heat transfer equation is a parabolic partial differential equation. We rewrite it in modem notation in the absence of a heat source and convective heat transfer  

Just as a random variable is characterized by the moments of its distribution, a stochastic process is characterized by its time correlation functions of various orders. In general, there are an infinite number of such functions, however we have seen that for the important class of Gaussian processes the first moments and the two-time coiTelation functions, simply referred to as time correlation functions, fully characterize the process. Another way to characterize a stationary stochastic process is by its spectral properties. This is the subject of this section. 

The nonlinearity of the current-voltage relationship in corroding systems provides an opportunity to determine corrosion rates without the need to measure independently the Tafel constants. The reason is that the electrical perturbation, which is imposed on the system at a frequency of /, in a nonlinear system results in a response at 2/, 3/, 4/, and so on, in addition to a dc component (McKubre [1983], Morring and Kies [1977], McKubre and Macdonald [1984], Bertocci [1979], Bertocci and Mullen [1981], Kruger [1903]). Neither the fundamental response (fo) nor the total power response can be analyzed to determine uniquely 

FIGURE 4.4.6.a. The source of a harmonic response reflection of an input sine wave on a 

The theoretical treatments referenced above all suffer from a major deficiency. The nonlinear term of interest in corrosion (the electron transfer process) is contained within a circuit comprising other linear (electrolyte resistance) and nonlinear (double-layer capacitance and diffusional impedance) terms. Since the voltage dropped across nonlinear circuit elements cannot be considered to linearly superimpose, we cannot use the equivalent circuit method to isolate the impedance terms of interest. Properly, one must solve for the system as a whole, including diffusional and double-layer terms, and identify the harmonic components associated with the faradic process of interest. 

The simplified theoretical treatment presented here is similar in form to that described previously (McKubre [1983], Bertocci [1979], Bertocci and Mullen [1981], Devay and Meszaros [1980], Devay [1982], Gill et al. [1983], Hladky et al. [1980], Rangarajan [1975], Ramamurthy and Rangarajan [1977], Rao and Mishra [1977], Callow et al. [1976], Devay and Meszaros [1980]). 

We are interested in the current response of an electrode to a voltage perturbation of the form 

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Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Harmonic analysis is an alternative approach to MD. The basic assumption is that the potential energy can be approximated by a sum of quadratic terms in displacements. 

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

Brooks, B. R., Janezic, D., Karplus, M. Harmonic Analysis of Large Systems I. Methodology. J. Comput. Chem. 16 (1995) 1522-1542 Janezic, D., Brooks, B. R. Harmonic Analysis of Large Systems II. Comparison of Different Protein Models. J. Comput. Chem. 16 (1995) 1543-1553 Janezic, D., Venable, R. M., Brooks, B. R. Harmonic Analysis of Large Systems. HI. Comparison with Molecular Dynamics. J. Comput. Chem. 16 (1995) 1554-1566... 

In an HT system, either the star is not grounded or it is a delta-connected system and hence the third harmonic is mostly absent, while the content of the. second harmonic nuiy be too small to be of any significance. For this purpose, where harmonic analysis is not possible, or for a new installation where the content of harmonies is not known, it is common practice to use a series reactor of 6% of the reactive value of the capacitors installed. This will suppress most of the harmonics by making the circuit inductive, up to almost the fourth harmonic, as derived subsequently. Where, however, second harmonics are significant, the circuit may be tuned for just below the second harmonic. To arrive at a more accurate choice of filters, it is better to conduct a harmonic analysis of the system through a harmonic analyser and ascertain the actual harmonic quantities and their magnitudes present in the system, and provide a correct series or parallel filter-circuits for each harmonic. 

But w hen the third and/or second harmonics are also present in the system, at a certain fault level it is possible that there may occur a parallel resonanee between the capacitor circuit and the inductance of the system (source), resulting in very heavy third or second harmonic resonant currents, which may cause failure of the series reactor as well as the capacitors. In such cases, a 6% reactor will not be relevant and a harmonic analysis will be mandatory to provide more exacting filter circuits. 

From the above it can be inferred that for an accurate analysis of a system, particularly where the loads are of varying nature or have non-linear characteristics it is necessary to conduct a harmonic analysis. The above corrective measures will provide a reasonably stable network, operat-ing at high p.f. with the harmonics greatly suppressed. The improved actual line loading, eliminating the fifth harmonic component, which is compensated,... 

Harmonic Analysis of Random Processes.—The response Y(t) of a linear, time-invariant electrical filter to an input X(t) can be expressed in the familiar form 66... 

N. Wiener, Generalized Harmonic Analysis, Acta Mathematical 55, 117-258 (1930) ... 

But this theory is rather difficult, and, as we shall see shortly, is not really necessary for the development of one of the most important parts of the theory of random harmonic analysis. 

Radiation field, interaction with nega-ton-positon field, 642 Radtke, M. ( ., 408 Raiffa, Howard, 314 Random coding, 227 Random particle velocity, 19 Random processes, 99,102 Gaussian, 176 harmonic analysis of, 180... 

Vibrational spectroscopy has played a very important role in the development of potential functions for molecular mechanics studies of proteins. Force constants which appear in the energy expressions are heavily parameterized from infrared and Raman studies of small model compounds. One approach to the interpretation of vibrational spectra for biopolymers has been a harmonic analysis whereby spectra are fit by geometry and/or force constant changes. There are a number of reasons for developing other approaches. The consistent force field (CFF) type potentials used in computer simulations are meant to model the motions of the atoms over a large ranee of conformations and, implicitly temperatures, without reparameterization. It is also desirable to develop a formalism for interpreting vibrational spectra which takes into account the variation in the conformations of the chromophore and surroundings which occur due to thermal motions. 

Brooks, B.R. lanezic, D. Karplus, M., Harmonic-analysis of large systems. 1. Methodology, 7. Comput. Chem. 1995,16, 1522-1542... 

The use of the impedance technique in the study of polymer coated steel, has been thoroughly described elsewhere. The present paper compares this technique with that of harmonic analysis, originally proposed by Meszaros ). The authors have presented preliminary data using the latter technique(3) wherein the early stages of polymer breakdown have been studied. The current paper extends this work to polymers which have been immersed for a considerable period of time. The harmonic method gives information not available from the impedance technique in the Tafel slopes and the corrosion current are directly measurable. A brief summary of the harmonic method and the equations used are given below. 

Harmonic analysis was carried out on the specimens 7 days after the impedance measurements in order to allow the specimens to settle down again. An Ono Sokki CF 910 dual channel FFT analyser was used in conjunction with a potentiostat (Thompson Mlnistat 251) to hold the specimen at its rest potential and to provide the low frequency sine wave perturbation. The second channel was used to measure the harmonic content of the resulting current. The Ono Sokki produces a dlgltially generated high purity sine wave at a chosen frequency, in this instance, 0.5 Hz. The total harmonic content of the input sine wave was less than 0.45% measured over 10 harmonics. Only the first 3 harmonics are used to calculate the corrosion current. 

As the impedance and harmonic analysis techniques gave different types of data from each other, a direct comparison between Tables 1 and 2 is difficult. However, it should be anoted that G4 gave both semicircular behaviour and a reasonably high corrosion rate. This similarity is also true for A14 and A15. All and A13 showed smaller but still measurable corrosion rates, together with semicircular impedance behaviour and the absolute values measured varied similarly as before. 

It can be seen that for severely degraded specimens, both the harmonic analysis and Impedance techniques are capable of detecting the presence of gross corrosion. The harmonics method provides a reasonable estimation of the corrosion rate when the Impedance data exhibits Warburg type behaviour. For less severely degraded specimens, especially those exhibiting blister attack, the Impedance method Is not as successful as the harmonic analysis technique. 

Where very little corrosion attack has occurred, neither method Is capable of providing reliable quantitative data. The non-appllcab-lllty In certain Instances of the Impedance technique as a monitoring tool has been reported previously. Further experience with the harmonic analysis technique may be capable of refining the results obtained. 

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