Reals. Axis

It remains to investigate the zeros of Cg t) arising from having divided out by. The position and number of these zeros depend only weakly on G, but depends markedly on the fomi that the time-dependent Hamiltonian H(x, () has. It can be shown that (again due to the smallness of ci,C2,...) these zeros are near the real axis. If the Hamiltonian can be represented by a small number of sinusoidal terms, then the number of fundamental roots will be small. However, in the t plane these will recur with a period characteristic of the periodicity of the Hamiltonian. These are relatively long periods compared to the recurrence period of the roots of the previous kind, which is characteristically shorter by a factor of... 

Symmetry of root loci The root loei are symmetrieal about the real axis. 

Asymptote intersection. The asymptotes interseet the real axis at a point given by... 

Root locus locations on real axis. A point on the real axis is part of the loei if the sum of the number of open-loop poles and zeros to the right of the point eoneerned is odd. 

Breakaway points The points at whieh a loeus breaks away from the real axis ean be ealeulated using one of two methods ... 

With the PD compensator of the form given in equation (5.98), the control problem, with reference, to Figure 5.14, is where to place the zero a on the real axis. Potential locations include ... 

Whichever physical interpretation is chosen, the difference between the high-frequency real axis intercept [Z (high) and the low-frequency limiting real impedance [Z (low)] is one-third of the film s ionic resistance (i.e., R[ = 3[Z (low) - Z (high)]). Ideally, the real component of the... 

Figures 5.29a and 5.29b show the Bode and Nyquist plot for a resistor, Ro, connected in series with a resistor, Rt, and capacitor, Ci, connected in parallel. This is the simplest model which can be used for a metal-solid electrolyte interface. Note in figure 5.29b how the first intersect of the semicircle with the real axis gives Ro and how the second intersect gives Ro+Rj. Also note how the capacitance, Ct, can be computed from the frequency value, fm, at the top of the semicircle (summit frequency), via C l JifmR .

For each k on the real axis, the sum over the occupation numbers nim diverges, so each integral should be taken before the summation. However, in the vicinity of the point that will turn out to be the saddle point k (Sfeo < all the sums are finite, so we reverse the order of summation and integration. The integration contour is shifted as shown in Fig. 16. 

Figure 16. The integration contour from Eq. (40) is distorted as explained in text. The pole k = iei is shown by a cross. Note the contour does not cross the pole when being shifted off the real axis.

Finally, for a complex pole, we can relate the damping ratio (t, < 1) with the angle that the pole makes with the real axis (Fig. 2.5). Taking the absolute values of the dimensions of the triangle, we can find... 

If we have chosen the other possibility of u = 0, meaning that the closed-loop poles are on the real axis, the ultimate gain is Kc u = -6, which is consistent with the other limit obtained with the Routh criterion. 

The root locus plot (Fig. E7.5) is simply a line on the real axis starting at s = -1/tp when Kc = 0, and extends to negative infinity as Kc approaches infinity. As we increase the proportional gain, the system response becomes faster. Would there be an upper limit in reality (Yes, saturation.)... 

If we increase further the value of Kg, the closed-loop poles will branch off (or breakaway) from the real axis and become two complex conjugates (Fig. E7.5). No matter how large Kc becomes, these two complex conjugates always have the same real part as given by the repeated root. Thus what we find are two vertical loci extending toward positive and negative infinity. In this analysis, we also see how as we increase Kc, the system changes from overdamped to become underdamped, but it is always stable. 

These are rough sketches of what you should obtain with MATLAB. The root locus of the system in (a) is a line on the real axis extending to negative infinity (Fig. E7.6a). The root loci in (b) approach each other (arrows not shown) on the real axis and then branch off toward infinity at 90°. The repeated roots in (c) simply branch off toward infinity. 

On the real axis, a root locus only exists to the left of an odd number of real poles and zeros. (The explanation of this point is on our Web Support.)... 

On the magnitude plot, the low frequency (also called zero frequency) asymptote is a horizontal line at Kp. On the phase angle plot, the low frequency asymptote is the 0° line. On the polar plot, the zero frequency limit is represented by the point Kp on the real axis. In the limit of high frequencies,... 

The results are exact—we do not need to make approximations as we had to with root locus or the Routh array. The magnitude plot is the same as the first order function, but the phase lag increases without bound due to the dead time contribution in the second term. We will see that this is a major contribution to instability. On the Nyquist plot, the G(jco) locus starts at Kp on the real axis and then "spirals" into the origin of the s-plane. 

On the negative real axis (-180°), find the "distance" of GCGP from (-1,0). This is the gain margin, GM. The formal definition is... 

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