This is an alternative analytical method in which the stability is assessed from an open-loop analysis.

The frequency response of the open-loop system G(s)H(s) is plotted in two distinct ways to give two equivalent stability criteria.

a. Nyquist criterion. This criterion allows one to calculate the gain and phase margins to give the degree of stability of the system, whereas the

Plotting Rules for Root-Locus Construction

Plotting rule

1 Number of locus branches equals order of denominator of G(s) H(s).

2 Locus is drawn for values of К as К varies from zero to infinity where К is

given by

G(s) H(s) = K(s-z)—/(s-p)—.

3 As К tends to infinity the locus of the closed-loop poles terminates on a zero

of G(s) H(s). Zeros at infinity are included. Also as К tends to zero the locus approaches the poles of G(s) H(s).

4 Loci on the real axis include those sections of the axis to the left of an odd

number of critical frequencies (zeros or poles) of G(s) H(s).

5 Loci near infinity asymptotically approach straight lines which meet the real

axis at an intersection point given by

sum(real part of poles) — sum(real part of zeros)
number of finite poles (p) — number of finite zeros (r)

and they meet the axis at angles given by в = плКр — z) for n odd.

6 Loci have symmetry about the real axis.

7 The values of К at which loci go into the right half-plane can be determined

graphically.

location of poles allowed one to obtain a rate of exponential decay that could be related to a damping factor as in the simulation technique of Section 2.5.2.1.

The frequency response of the open-loop system G(s)H(s) is plotted on an Argand diagram and the criterion states: “If a system is unstable the number of unstable frequencies is given by the number of times this fre­quency response curve encircles the —1 point in a clockwise direction.” Defining

W{s) = G(s)H(s) + 1 (2.17)

to be the closed-loop characteristic, from Cauchy’s theorem in complex number theory, assuming that IT(.s) is single-valued and has finite poles in a contour C and is not zero on C, then

Jc W'{s)IW{s) = 2ni(Z — P)

[In W(s)]c = 2ni(Z — P)

= A2 + iB2 — Ax — iBx where Fig. 2.29a defines the anticlockwise path around contour C.

Fig. 2.29a. C contour in the і plane.

Аг — A2

В2— B1 = 2n(Z — P)

One revolution of the W plane origin gives Ini, thus the number of encir­clements of the W origin is equal to Z — P.

The stability criterion now arises because the closed-loop transfer func­tion pole is a zero of W(s) and therefore for stability there should be no zeros in the right half-plane of W(s). For stability there must be as many encirclements as W(s) has poles in the right half-plane.

Thus the number of encirclements of the origin in the W(s) plane (Z — 0) is equal to the number of encirclements of the point —1 in the G(s)H(s) plane, and this in turn is equal to the number of poles of the open-loop transfer function as shown above. The negative sign now implies clockwise encirclements. Examples are given in Section 2.5.3.3.

The effect of a time lag is to make the closed-loop system more unstable in general. See the second Nyquist example in Section 2.5.3.3.

b. Bode plot method. Again using the open-loop response of the system, the phase and gain may be plotted and the Nyquist criterion used in a slightly different manner.

The gain is defined as M where:

M = 20 log10 (response argument)

and the gain and phase are plotted against frequency. Because of the loga­rithmic formulation the gains and phases can be added for multiplicative systems, so that the frequency responses of sections of the system can be easily totalled. See the example in Section 2.5.3.3.

Thus the response of the neutron kinetics can be calculated separately from the thermal characteristic response, and the feedback and each can be converted into a separate tabulation of gain and phase versus frequency which can be summed. The only diffculty occurs where multiple feedback paths exist.

The Nyquist diagram for the — 1 point is translated here into a similar one on the Bode diagram for the point of 0 dB and a phase of —180°. For mere stability, the system must have a negative gain at 180° but, in practice, for suitable operational stability one needs some damping, which is provided by a gain margin of about 20 dB (Fig. 2.29b).