This method does exactly what it states, it locates the poles by solving the characteristic equation from the closed-loop transfer function:

1 + G(s)H(s) = 0 (2.12)

In a reactor system, these poles will be functions of core parameters, the power level, and plant time constants.

Depending on the degree of s in G(s)H(s), various criteria can be esta­blished for defining the stability of the system. These are algebraic condi­tions on the coefficients of Eq. (2.12) to ensure that there are no poles in the right half of the complex s plane.

a. Routh-Hurwitz method. This method (7) starts from the closed-loop characteristic [Eq. (2.12)], written as

ansn + an_i5n_1 + • • • + a0 = 0 (2.13)

Then an array of n + 1 rows is prepared as follows:

(2.14)

Ci = (Vn-з — I bi, etc.

The number of roots is the number of sign changes in the first column of this array (2.14). Thus the stability criterion is that there should be no changes of sign in this column and thus no poles in Eq. (2.11).

b. Root-Locus method. This method also ensures no poles in the right half of the plane by drawing a locus of values of s which satisfies the char­acteristic Eq. (2.12). The locus is drawn by a graphical method and then the stability criterion is stated in terms of the points at which the locus passes into the right half-plane. A knowledge of G(s)H(s) is required as well as its zeros and poles and the locus is drawn from these points such that a value of s on the locus satisfies the two equations

I G(s)H(s) I = 1 arg G(s)H(s) = nn

which is Eq. (2.12) in its gain and phase components. The rules for drawing the root-locus and the statement of the criteria for stability are summarized in Table 2.2. The subject is treated in excellent fashion by Weaver (7).

c. Solution of the characteristic equation. If the system is simple, then one final method is available. The characteristic equation could be solved for its roots [that is, for the poles of Eq. (2.11)], given all the values of the relevant coefficients, which would include heat-transfer coefficients, feed­back coefficients, and time constants. In even a simple system, this method can be, at best, time-consuming; in a more complex system it is generally impossible.