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A number of other methods of determining system stability—such as the i Nyquist criterion, and root locus diagrams—have been developed in the last three decades (see, for example, Melsa and Shultz  and Greensite ). The most common of these is the root locus diagram, which is a plot in the complex s-plane of all possible locations of the roots of the characteristic equation of the system's ,■ i closed-loop transfer function as the gain, K, is increased from zero to infinity.

Let GH, the open-loop transfer function, be represented as the ratio of two _ j

i mi polynomials in s:

where K is the system gain. Then the closed-loop transfer function is c G(s) G(s) = </(*)(?(*)

The poles of the closed-loop transfer function are the roots of d(s)+ Kn(s)=0. As the value of K changes, the location of these roots in the complex s-plane also changes. A root locus diagram is the locus of these roots as a function of K. The locus of a particular root is a branch on the root locus diagram. For K=0, the roots of the characteristic equation are the roots of rf(j)=0, that is, the poles of the open-loop transfer function. As K increases from zero to infinity, these roots approach the roots of n(s), i.e., the zeros of the open-loop transfer function. Therefore, as the value of K increases from zero to infinity, the loci of the poles of the closed-loop transfer function start at the open-loop poles and terminate at the open-loop zeros. If, for a given K, none of the roots of the characteristic equation has positive real parts, then the system is stable.

A set of general rules for constructing and interpreting root locus diagrams follows:

1. The number of loci, or branches of the root locus, is equal to the number of poles of the open-loop transfer function, GH= KB(s).

2. The root loci are continuous curves. The slopes of the root loci are also continuous except for points at which either dB(s)/ds = Q, K=0, or B(s) is infinite.

3. Loci begin at poles of B(s) where K=0, and terminate at zeros of B(s), where K is infinite.

4. If the open-loop transfer function, KB(s), has p finite poles and z finite zeros, there will also bep— z zeros at infinity if p>z.

5. For a branch of the root locus diagram to pass through a particular value of i—say, i,—S| must be one of the roots of the characteristic equation d(s,)+Kn(s,) =0 for some real value of K. The condition for which s, is the root of the characteristic equation is that B(s,) must have a phase angle and magnitude given by argS(j|)=((2/+1),rradianS' *>0 (18-34)

where / is an arbitrary integer. To satisfy Eq. (18-34), the magnitude of KB(st) must be equal to unity, and its associated phase angle must be an odd multiple of it radians. These two criteria are known as magnitude and angle criteria, respectively.

6. Branches of the root locus are symmetrical with respect to the real axis because all complex roots appear in conjugate pairs.

7. For p> z and |j|s>0, branches of the root locus approach a set of asymptotic straight lines. The asymptotes to the loci at infinity meet at the centroid of B(s) given by p i 2 P~ 2

where p, and z, denote the /th pole and zero of KB(s). The angles between the asymptotes and the real axis are