Even harmonics

The third harmonic was almost absent, hence it was not considered. The second and other even harmonics were also insignificant. Similarly, higher harmonics (above the thirleenth) were also insignificant, hence were not considered for ease of illustration. 

So far, we have not included even harmonics in the discussion doing so is left to the reader as an exercise. Table 4.1 categorizes the harmonics in terms of their respective sequence orders. 

Figure 4.10 shows a current waveform at a distribution panel supplying exclusively fluorescent lights. The waveform is primarily comprised of the third and the fifth harmonic frequencies. The individual current harmonic distortion makeup is provided in Table 4.2. The waveform also contains slight traces of even harmonics, especially of the higher frequency order. The current waveform is flat topped due to initiation of arc within the gas tube, which causes the voltage across the tube and the current to become essentially unchanged for a portion of each half of a cycle. 

Note Isc = maximum short-circuit current at PCC IL = maximum fundamental frequency demand load current at PCC (average current of the maximum demand for the preceding 12 months) h = individual harmonic order THD = total harmonic distortion, based on the maximum demand load current. The table applies to odd harmonics even harmonics are limited to 25% of the odd harmonic limits shown above. 

Another more sophisticated approach is to make a Fourier Transform analysis of the response in the way proposed by Bond et al. [84, 85]. In this case, the perturbation is a continuous function of time (a ramped square wave waveform) which combines a dc potential ramp with a square wave of potential that can be described as a combination of sinusoidal functions. Under these conditions, the faradaic contribution to the response generates even harmonics only (i.e., the non-faradaic current goes exclusively through odd harmonics). Thus, the analysis of the even harmonics will provide excellent faradaic-to-non-faradaic current ratios. 

The inherent feature of the quadratic SR under study is that the bias field is the sole cause of even harmonics in the spectrum. Due to the symmetry considerations, they must vanish at H —> 0. This means that S2 (E, —> 0) = 0. In Eq. (4.290) this limit is ensured by the proportionality of S2 to B. According to the second of Eqs. (4.253), the coefficients consists only of the odd equilibrium moments of the distribution (4.226). Since the function Wo is even in x at E, = 0, the odd moments vanish. 

Calculate the value of X. Determine b using X (better use X obtained from (20) for even harmonic). Obtain, s- using formula (18). 

Fig. 9. Difference spectrum of metabolising yeast cells Even harmonics appear under these conditions showing an activation of the ATPase signifying the disturbance of the equilibrium of Fig. 5...

Despite the powerful symmetry rule that precludes the generation of even harmonics in optically isotropic media, except at surfaces, a number of experimental results have indicated exceptions to the rule, as detailed in the earlier review [1]. Most entail conditions resulting in a transient, local removal of isotropy, and are therefore well understood. Nonetheless, two quite different mechanisms have been found to mediate second-harmonic generation in macroscopically isotropic systems. In this section we consider a mechanism relating to optical coherence in small particles in suspension, or locally ordered domains within macroscopically structureless media. In the next section we shall focus on a six-wave form of interaction associated with very high pump laser intensities. 

Odd harmonics (n = 1,3) are ideally on-axis, and even ones are off-axis. For a real ring with non-zero emittance, however, the phase envelopes of the harmonics overlap, and all are accessible on-axis. The formal bandwidth of each harmonic wavelength is 1 /tiN, but deviation of the magnetic field from ideal causes broadening of the peaks. In general, the odd harmonics have linear polarization on-axis, while the polarization of the even harmonics is complex. 

The series in (4) and (5) are known as odd-harmonic series, since only the odd harmonics appear. Similar rules may be stated for even-harmonic series, but when a series appears in the even-harmonic form, it means that 2L has not been taken as the smallest period of f(x). Since any integral multiple of a period is also a period, series obtained in this way will also work, but in general computation is simplified if 2L is taken to be the smallest period.)... 

It is found from the integration that all even harmonics and those multiples of three are not present in the waveform. Hence n has the following value for a six-phase bridge ... 

The 180° waveform contains triplen harmonics for n taking odd values. The 60° waveform also contains the same triplen harmonics but with opposite signs, which therefore cancel those in the 180° waveform. None of the waveforms contain even harmonics. 

The structure of the formalism of fhe MEMP theory outlined above, allows the easy computation from firsf principles of the effects of polychromatic fields wifh commensurafe frequencies. This fype of information is useful for fhe quanfifafive undersfanding of possibilities of control of ionization as a function of fhe paramefers of fhe fields. For example, for a weak dichromatic field, we have demonsfrafed fhaf fhe ionization rate, P, has the following dependence on fhe relafive phase y [188, 189] and references therein. For even harmonics, = 2k)co2,... 

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