Several examples are presented to clarify certain sections of the foregoing text

a. Routh-Hurwitz criterion (refer to Section 2.5.3.1). Figure 2.30 shows a simple feedback loop for an imaginary fluid-fueled system in which the fuel produces a power P while in the critical configuration of the core but gives up a proportion of its heat a while on a path through an IHX. The circuit time is в. The equations are:

Fuel

dT/dt = P— h(T— Ті)

Neutron kinetics

dPjdt = (6k PIl*) — (aTjl*) + (aTjl*)

Circuit

Ti = aT with a delay of 6

No delayed neutrons are assumed. Taking the Laplace transform,

ST — T0 = P — hT + hT{ (2.27)

sP-P0= (6k PJl*) + (6k0 P/i*) — (aTjl*) (2.28)

= «77(1 + 6s) (2.29)

and in the steady state it is assumed that

Ti0 = aT0 and 6k0 == 0

P0 = h(T0 — Ti0) and take P0 = 1 (2.30)

From these relations the frequency response of each pair of variables Tj6k, P/6k, and TJdk may be calculated. Defining G(s) = P/6k" and H(s) = 6k’ jP, where

6k" = 6k—6k’ and 6k’ = (aT—aT0) (2.31)

the characteristic equation is given by

so 1 + G{s)H{s) — 0 gives a cubic equation:

s3l*6 + s4*( 1 + 6h) + s(/*A(l + a) + «0) + a = 0 (2.34)

The Routh-Hurwitz criterion for stability gives the following array:

1*6 l*h( + a) + ad

/*(1 + Bh) a

had2 _ (2.35)

(1 + 6h)

0

from which the criterion of no sign changes in the first column gives:

Both of these statements are satisfied by a positive a, so the system is stable for all negative Doppler coefficients (—a).

b. Nyquist criterion. (1) A system in which G(s) = —K(s + 2)_1 is shown in Fig. 2.31. To draw the locus of G{im) notice that for

(o = 0, G(ico) = —K

(o = oo, G(im) = 0

Thus the locus starts from the origin and has a diameter of K (Fig. 2.32). The number of poles P of G(s) in the right half-plane is zero (only s = — 2 exists in the left half-plane) so, for stability (Z = 0), there should be no encirclements of the origin by 1 + G(s). To avoid any encirclements by G(s) of the point — 1 and to ensure stability for this system, the criterion is that К must be less than 2. Any adjustment to the system to make К smaller to obtain this condition will be a gain adjustment.

Fig. 2.32. Nyquist plot (7) for G(/co).

(2) If the system is now represented by

G(s) = Ks-^s + a)-1^ + b)-1 (2.38)

with a feedback loop shown in Fig. 2.33, the locus in the G(ico) plane is shown in Fig. 2.34a as for

a> = 0, G(ico) = oo

со = go, G(ico) -= 0

The plot in the s plane is shown in Fig. 2.34b. Here P = 0 for a, b > 0 so there should be no encirclements. The stability criterion is now that K/ab(a + b) must be less than unity to avoid encirclements of the point — 1. Thus the maximum gain К is ab(a + b). This is called the gain margin when related to the actual value of K.

Note that in this example an wth order pole at the zero point in the s plane is mapped into a counterclockwise rotation of rm of infinite mag­nitude in the G{s) plane. H(s) is, of course, unity in these examples.

(3) The effect of a time lag is that it tends in general to make the closed — loop system more unstable. The lag exp(—sT) in association with G(s)=K/s

Fig. 2.34a. The s plane contour (7).

Fig. 2.34b. Nyquist diagram (7) for G(iw) = Kl[io)(ia) + a)(iw + b)].

gives the Nyquist diagram shown in Fig. 2.35, where the time lag is a linearly increasing, frequency-dependent function. For this system, the stability criterion is that К should be less than nT.

c. Bode plot. For a system characterized by

G(s) = K/(l + sT)» (2.40)

the gain is

20 log] G{i<o) I = 20 log К — n20 log] (1 + іТш) | (2.41)

The separate parts of this expression can be plotted and added as shown in Fig. 2.36.

Fig. 2.36. Log magnitude as a function of the log of the frequency.

If now a feedback H(s) = F(1 + s9) is required, then the gain of G(s)H(s) is given by the sum

total gain = 20 log AT — 20nlog|(l + гТсо)| + 201ogF + 20 log] (1 + івш)|

(2.42)

for all frequencies со. In order to establish the stability margin, this gain is plotted against the phase after eliminating the frequency parameter. The criterion of Section 2.5.3.2 is used.