Nyquist equation

The sampling time during which FID data points must be collected in order to obtain the true NMR spectrum after Fourier transformation depends on the spectral width A. According to information theory [18], the sweep time per data point, called the dwell time faw, must satisfy the minimum given by the Nyquist equation (2.12). 

A frequency domain stability criterion developed by Nyquist (1932) is based upon Cauchy s theorem. If the function F(s) is in fact the characteristic equation of a closed-loop control system, then... 

Then n in equation (6.62) is the type number of the system and J([ denotes the product of the factors. The system type can be observed from the starting point uj 0) of the Nyquist diagram, and the system order from the finishing point bj oo), see Figure 6.22. 

Alternatively, the closed-loop frequency response can be obtained from a Nyquist diagram using the direct construction method shown in Figure 6.25. From equation (6.73)... 

Consider a Nyquist contour for the nominal open-loop system Gm(iLu)C(iuj) with the model uncertainty given by equation (9.119). Let fa( ) be the bound of additive uncertainty and therefore be the radius of a disk superimposed upon the nominal Nyquist contour. This means that G(iuj) lies within a family of plants 7r(C(ja ) e tt) described by the disk, defined mathematically as... 

From the Nyquist stability criterion, let N k, G(iuj)) be the net number of clockwise encirclements of a point (k, 0) of the Nyquist contour. Assume that all plants in the family tt, expressed in equation (9.132) have the same number ( ) of right-hand plane (RHP) poles. 

An important early paper on fluctuation processes is that of Harry Nyquist (1928), who suggested an equation linking the mean-square amplitude of thermal noise in an electrical circuit to the resistance R of the noise EMF (or current) generator ... 

A preview We can derive the ultimate gain and ultimate period (or frequency) with stability analyses. In Chapter 7, we use the substitution s = jco in the closed-loop characteristic equation. In Chapter 8, we make use of what is called the Nyquist stability criterion and Bode plots. 

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

Nyquist stability criterion Given the closed-loop equation 1 + Gol (joi) = 0, if the function G0l(J ) has P open-loop poles and if the polar plot of GOL(](o) encircles the (-1,0) point... 

Once we understand the origin of Nyquist stability criterion, putting it to use is easy. Suppose we have a closed-loop system with characteristic equation 1 + GCGP = 0. With the point (-1,0) as a reference and the Gc(jco)Gp(jco) curve on a... 

The Nyquist stability criterion is, on the surface, quite remarkable. We are able to deduce something about the stability of the closedloop system by making a frequency response plot of the openloop system And the encirclement of the mystical, magical (— 1, 0) point somehow tells us that the system is closedloop unstable. This all looks like blue smoke and mirrors However, as we will prove below, it all goes back to finding out if there are any roots of the closedloop characteristic equation in the RHP. 

All the Nyquist, Bode, and Nichols plots discussed in previous sections have been for openloop system transfer functions B(j ). Frequency-response plots can be made for any type of system, openloop or closedloop. The two closedloop transfer functions that we derived in Chap. 10 show how the output is affected in a closedloop system by a setpoint input and by a load. Equation (13.28) gives the closedloop servo transfer function. Equation (13.29) gives the closedloop load transfer function. 

The Nyquist stabihty criterion can be used for openloop unstable processes, but we have to use the complete, rigorous version with P (the number of poles of the closedloop characteristic equation in the RHP) no longer equal to zero. 

Sampled-data control systems can be designed in the frequency domain by using the same techniques that we employed for continuous systems. The Nyquist stability criterion is applied to the appropriate closedloop characteristic equation to find the number of zeros outside the unit circle. 

As we will show in a minute when we have completed the rest of the contours, this means that if the controller gain is made big enough, the Nyquist plot will encircle the ( — 1, 0) point. If IV = 1, Z = 1 for this system since P = 0. Thus there will be one zero or root of the closedloop characteristic equation outside the unit circle. 

There is an alternative way to generate the Nyquist plots that is often more convenient to use, particularly in high-order systems. Equation (18.13) gives a doubly infinite series representation of. ... 

The SIN defined by Equation 7.6 for a given NMR resonance is proportional to the square of the nuclear precession frequency (mo, rad/s), the magnitude of the transverse magnetic field (Bi) induced in the RE coil per unit current (/), the number of spins per unit volume (Ns), the sample volume (Vs), and a scaling constant that accounts for magnetic field inhomogeneities. The SIN is inversely proportional to the noise generated in the RE receiver and by the sample (Vnoise) as defined by the Nyquist theorem,... 

It can be shown061 that, if there are any net encirclements of the point (-1,0) on the Nyquist diagram (i.e. if nE > 0), then the system characteristic equation will have roots lying to the right of the imaginary axis and consequently the system will be unstable (Fig. 7.53). 

From example 7.6 we know that critical stability occurs for Ac = 1.8, r, = 3.5. Hence, by the Nyquist criterion, when these conditions are applied, the polar plot will pass through the point (-1,0) on the complex plane, i.e. for these values of the controller parameters, 9m (G(i[Pg.631]

It can be shown that there is a Nyquist criterion for sampled data systems which is equivalent to that for continuous systems (see Section 7.10.5) and equation 7.131 can be applied in its comparable r-transformed form(42). In practice it is generally sufficient to ascertain whether the polar plot of G(z) in the complex z-plane encircles the (-1,0) point (as with continuous systems in the j-plane) where 1 + G(r) = 0 is the system z-transformed characteristic equation. The polar plot is constructed from... 

Equation (8.6) is a simplified approximation. For an exact definition, please see, for example, Bard and Faulkner (2001). It is mentioned here because it appears at low frequency (0-20 Hz) excitations. At higher (>100Hz) frequencies, the second term can be neglected and the Nyquist plot reverts to the simple semicircle. 

Equations 2.37-2.40 result in the commonly used presentation of the impedance, e.g. the Nyquist and the Bode plots. The first one shows the total impedance vector point for different values of co. The plane of this figure is a complex plane, as shown in the previous section. Electrochemical-related processes and effects result in resistive and capacitive behaviour, so it is common to present the impedance as ... 

It is often experimentally observed that in the Nyquist plot, semi-circles are obtained with a centre point below the x-axis. Analysis of the situation tells us that the double-layer capacity is not a suitable description of the system occurring and should be replaced by an element with an impedance function given in Equation 2.51 ... 

In the plot in Figure 8.23, it is evident that the ohmic factors are independent of frequency, after this, the ideal activation processes display a semicircular conduct with a frequency which is typical of the corresponding relaxation processes (see Equation 8.88 and Figures 8.20 and 8.21) finally, the concentration processes exhibit a diagonal conduct characteristic of diffusion processes (see Figure 8.22) often referred to as the Warburg behavior [124,129,130] (to see a real Nyquist plot related to an EIS test of a battery, see Section 8.9.1). 

Table 4. Summary of values of it constant (see Equation 1) for the most important substituents present in chemicals relevant to the CWC (see Thomas <5) for condensed phase values and Nyquist<45) for vapor phase values)...

From Equation 4.31, the simulated Nyquist plot is presented in Figure 4.7b. 

Interestingly, due to the linearity of the generalized Langevin equation (22), the same effective temperature T,eff(( ) can consistently be used in the modified Nyquist formula linking the noise spectral density C/ /- ([Pg.313]

Nemsi diffusion layer, 5, 57, 86, 98, 352, 451 Nemst equation, 8, 21, 101, 138, 147, 381 Nickel-cadmium baiteries,473 Nonisothermal enthalpy of activation, 153 Nonpolarizable interphase, 8 33 Normal hydrogen electrode scale, 27 Numerical value of b, 123 Nyquist plot, 215... 

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