Crossover frequency

It is observed that Re(fcM), that is, the real part of the Clausius-Mossoti factor determines the direction of the DEP force. Let us investigate the Clausius-Mossoti factor as a function of the electric field frequency. 

The real part of the Clausius-Mossoti factor is written as 

Note that Reifcu) varies between two extremum values of frequency  

Note that at low frequencies ( 10 kHz), Cp is the function of the frequency of the electric field. This is attributed to the relaxation of the double layer surrounding the particle, which depends on the radius of the particle with respect to the Debye length. This situation is complex, and no satisfactory model exists to describe this phenomenon. The counterions of the double layer do not have enough time to move in case of high frequencies. Thus, at high frequencies, the particle is nondispersive compared to that at low frequencies. Hence, Cp and Tp are independent of frequency at high frequencies. Here, the polarization is primarily due to the particle with respect to the surrounding medium. The behavior of the particles switches 

The crossover frequencies from the above expression have shown some discrepancy with experimental observation. Therefore, some authors have considered this problem by assuming an infinitely thin conductive layer on the surface of the particle due to the double layer, leading to modification of the conductivity of the particle as 

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The gain eross-over frequency of the closed loop should not be any higher than 20 percent of the switching frequency (or 20 kHz). I have found that gain crossover frequencies of 10 kHz to 15 kHz are quite satisfactory for the majority of applications. This yields a transient response time around 200 uS. 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

Place ujm at the modulus crossover frequency of 2 rad/s and position the compensator corner frequencies an octave below, and an octave above this frequency. Set the compensator gain to unity. Flence... 

Figure 6.35 shows the Bode gain and phase for both compensated and uncompensated systems. From Figure 6.35, it can be seen that by reducing the open-loop gain by 5.4dB, the original modulus crossover frequency, where the phase advance is a maximum, can be attained. 

Required modulus attenuation is 12 dB. This reduces the modulus crossover frequency from 1.4 to 0.6 rad/s. 

Find the frequency where the magnitude Gc(jto)Gp(jco) is 1. This particular frequency is the phase crossover frequency, to = tocp. We then find the angle between GcGp and -180°. This is the phase margin, PM. The formal definition is... 

To find the new gain margin, we need to, in theoiy, reverse the calculation sequence. We first use the phase equation to find the new crossover frequency (Dcg. Then we use the magnitude equation to find the new GOL, and the new GM is of course 1/ G0lI However, since we now know the values of td, xp, and KcKvKpKm, we might as well use MATLAB. These are the statements ... 

Frequency methods can give us the relative stability (the gain and phase margins). In addition, we could construct the Bode plot with experimental data using a sinusoidal or pulse input, i.e., the subsequent design does not need a (theoretical) model. If we do have a model, the data can be used to verify the model. However, there are systems which have more than one crossover frequency on the Bode plot (the magnitude and phase lag do not decrease monotonically with frequency), and it would be hard to judge which is the appropriate one with the Bode plot alone. 

We now find the gain margin with its crossover frequency (Gm, Wcg), and phase margin with its crossover frequency (Pm, wcp) with either one of the following options ... 

The recombination frequency provides a measure of the genetic distance between any pair of linked loci. Genetic distances are often expressed in centiMorgans (cM). One centiMorgan is equal to a 1% recombination frequency between two loci (for example, two loci that are 10 cM apart would have a recombination frequency of 10%). Physically, 1 cM is approximately equal to 1 million base pairs of DNA (1 Mb). This relationship is only approximate, however, because crossover frequencies vary somewhat throughout the genome (e.g., crossovers are less common near centromeres and more common near telomeres). 

Various factors govern autohesive tack, such as relaxation times (x) and monomer friction coefficient (Co) and have been estimated from the different crossover frequencies in the DMA frequency sweep master curves (as shown in Fig. 22a, b). The self-diffusion coefficient (D) of the samples has been calculated from the terminal relaxation time, xte, which is also called as the reptation time, xrep The D value has been calculated using the following equation ... 

With the above boundary conditions, we can calculate a general surface mode of uniaxial gels. However, we restrict ourselves to a slowly diffusing mode (oc exp(— Dsk2t)) with its decay rate much smaller than the crossover frequency (oc given by Eq. (6.21). After some calculations, the diffusion constant Ds of this mode is obtained as [92]... 

It is also important to optimize the length of the annealing/extension step. You should start with a shorter annealing/extension time. Because of the very fast polymerase activity, very often full-length product can be achieved after only 10-20 cycles, but with lower crossover frequency. 

The reduced background of unshuffled clones is a significant advantage with these techniques, with over 75% shuffled clones in most cases and 100% with RACHITT. In addition, the higher crossover frequency observed with RACHITT, and the average of 14 per gene compared to one to four for other PCR-based methods, leads to a considerable increase in library diversity (Pelletier, 2001). 

Fig. 10. Modifying the resonances in an electrode system can be useful when two cell types have only small differences in passive electrical properties and in dielectrophoretic force spectra. The best frequency to achieve a separation lies near the crossover frequencies when one cell is showing positive DEP and the other negative. But at such a frequency the forces are small. By suitable adjustment of capacitive and inductive elements at each electrode, it is possible to make a system resonate at the desired frequency, thereby increasing the drive voltage (and force) many fold without the need for expensive high voltage signal generators. The real dielectrophoretic force spectra (a) can be transformed into effective spectra (b)...

The parameters to be optimized for experimental recombination include the crossover frequency, the number of parents, and the sequence similarity between parents. Additionally, it is important to understand the conditions under which recombination is useful. For all these questions, the optimal parameters represent a balance between the exploration and exploitation capabilities of the search algorithm. Any process that creates more diversity, such as using many parents, very disparate parents, or small fragment sizes, will improve exploration at the cost of exploitation. The balance between these effects will shift according to the landscape ruggedness and the sampling ability of the mutant library. 

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