Harmonic components

From this it can be seen that vibration is the universal manifestation that something is wrong. Therefore, many units are equipped with instruments that continuously monitor vibration. Numerous new instruments for vibration analysis have become available. Frequency can be accurately determined and compared with computations, and by means of oscilloscopes the waveform and its harmonic components can be analyzed. Such equipment is a great help in diagnosing a source of trouble. 

Such relays are normally instantaneous, highly sensitive and operate at low spill cuiTents. Since they detect the residual current of the system, the current may contain third-harmonic components (Section 23.6) and operate the highly sensitive relay in a healthy condition. To avoid operation of the relay under such conditions, it is a normal practice to supply the relay coil with a tuned filter, i.e. a series L-C circuit to filter out the third-harmonic components. The capacitance of the filter circuit may also tame a steep rising TRV (Section 17.10.3) during a momentary transient condition and protect the relay. 

Note t, would consist of resistive as well as 3rd harmonic component. 

Current probe - to measure the third harmonic component of/p It is then converted to actual I, from the ZnO characteristic data provided by the arrester manufacturer, /r versus /jr, corrected to the site operating temperature and voltage. The value of is then used to assess the condition of the arrester. 

A harmonic component affects the performance of a capiicitor unit significantly due to diminishing reactance at higher frequencies, which adds to its loading substantially and can be analysed as follows ... 

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,... 

The French physicist and mathematician Jean Fourier determined that non-harmonic data functions such as the time-domain vibration profile are the mathematical sum of simple harmonic functions. The dashed-line curves in Figure 43.4 represent discrete harmonic components of the total, or summed, non-harmonic curve represented by the solid line. 

In most cases, the half-harmonic components are about one-half of the amplitude of the harmonic components. They result from the machine-train lifting until stopped by the bolts. The impact as the machine reaches the upper limit of travel generates a frequency component at one-half multiples (i.e., orders) of running speed. As the machine returns to the bottom of its movement, its original position, a larger impact occurs that generates the full harmonics of running speed. 

When compared to standard (open cavity) cone-plate or parallel disks rheometers, closed cavity torsional rheometers such as the RPA or the PPA have unique high-strain capabilities, which prompted us to modify the instmment in order to investigate the promises of FT rheometry, as outlined a few years ago by the pioneering works of Wilhelm. The technique consists of capturing strain and torque signals and in using FT calculation algorithms to resolve it into their harmonic components, as detailed below. 

FT is essentially a mathematical treatment of harmonic signals that resolved the information gathered in the time domain into a representation of the measured material property in the frequency domain, as a spectrum of harmonic components. If the response of the material was strictly linear, then the torque signal would be a simple sinusoid and the torque spectrum reduced to a single peak at the applied frequency, for instance 1 Hz, in the case of the experiments displayed in the figure. A nonlinear response is thus characterized by a number of additional peaks at odd multiples of the... 

FIGURE 30.S Fourier transform (FT) resolution of torque signals into harmonic components notes that relative harmonic components are displayed. 

Fourier transform spectrum odd-harmonic components analysis... 

According to test protocols described above, RPA-FT test were performed at 100°C, 1 Hz on all samples additional tests at 100°C, 0.5 Hz were performed on IMA TR, IMA FM, and IMA-AG samples. Essentially three types of data will be discussed hereafter The complex modulus G (as derived from the main torque component in the FT torque spectrum), the corrected total torque harmonic component, i.e., cTTHC, and the Q1/Q2 ratio. 

FIGURE 30.15 Ethylene-propylene-diene monomer (EPDM) compounding corrected relative 3rd torque harmonic component and total torque harmonic content (TTHC) versus strain, at 0.5 Hz samples TR, EM, and AG. 

The second equality in Eq. (4.3.2) demonstrates that the harmonic component of the Hamiltonian of the molecular subsystem is diagonalized by the Fourier transform in terms of wave vectors K of vibrational excitations ... 

Figure 18A shows the Fourier spectra thus obtained. The real and imaginary parts correspond to the elastic and viscous components of the DOPC thin film, respectively. We can see that the spectrum is composed not only from the fundamental (coo) but also from the higher (2thin film exhibits rather large nonlinearity in the viscoelastic characteristics. 

The factor f reduces the oscillation amplitude symmetrically about R - R0, facilitating straightforward calculation of polymer refractive index from quantities measured directly from the waveform (3,). When r12 is not small, as in the plasma etching of thin polymer films, the first order power series approximation is inadequate. For example, for a plasma/poly(methyl-methacrylate)/silicon system, r12 = -0.196 and r23 = -0.442. The waveform for a uniformly etching film is no longer purely sinusoidal in time but contains other harmonic components. In addition, amplitude reduction through the f factor does not preserve the vertical median R0 making the film refractive index calculation non-trivial. 

Fig. 15 Raman scattering spectra of a STO 16 and b STO 18-23 observed in x(yy)-x scattering geometry (tetragonal notation). Arrows in a indicate positions of DIRS signal and its higher harmonic component [27]...

Harmonic Frequency. A Vibrational Frequency which has been corrected to remove all non-quadratic (non-harmonic) components. Calculated Vibrational Frequencies correspond to harmonic frequencies. The corrections require data on isotopically-substituted systems and are typically available only for small molecules. 

The tip wavefunctions can be expanded into the spherical-harmonic components, T/, (0, ( )), with the nucleus of the apex atom as the origin. Each component is characterized by quantum numbers I and m. In other words, we are looking for solutions of Eq. (3.2) in the form... 

A disadvantage of the system described in the preceding paragraphs is that, in general, the time-harmonic components of the signal contain mixtures of... 

In WMS, phase-sensitive (i.e. lock-in) amplification demodulates the periodic signal with a very narrow electrical bandwidth to precisely measure the RMS amplitudes of the fundamental sinusoid and its second harmonic component, averaged over a period of time equal to the inverse of the electrical bandwidth. These are called the If and 2/ signals and each amplitude is expressed by the following equations, respectively,... 

The motivation for the sine-wave representation is that the waveform, when perfectly periodic, can be represented by a Fourier series decomposition in which each harmonic component of this decomposition corresponds to a single sine wave. More generally, the sine waves in the model will be aharmonic as when periodicity is not exact or turbulence and transients are present, and is given by... 

In adding synthetic harmonic Mid aharmonic components, it is important that die two components fuse perceptually [Hermes, 1991] i.e., that the two components are perceived as emanating from the same sound source. Through informal listening, sine-wave phase randomization appears to yield a noise component that fuses with the harmonic component of the signal. 

The significance of harmonic frequencies can be seen in Figure 4.3. The second harmonic undergoes two complete cycles during one cycle of the fundamental frequency, and the third harmonic traverses three complete cycles during one cycle of the fundamental frequency. Vx, V2, and V3 are the peak values of the harmonic components that comprise the composite waveform, which also has a frequency of/. 

---