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For small signal stability analysis, eigenvalue is calculated from system characteristic equation:

(1.54)

where unknown root λ is calculated (i.e., eigenvalue) with A and I as system matrix and identity matrix, respectively. For a system to be stable, all the real and/or real component of eigenvalues must be negative [15, 18–20].

State equation (1.53) and equation 8.3-45 of reference [15] are using nine and seven state variables, respectively. Hence, nine and seven eigenvalues will be obtained for six and three-phase generator, respectively. In six-phase generator, out of nine evaluated eigenvalues, three eigenvalues are complex conjugate pairs and the remaining are real. Evaluated eigenvalue of six-phase and three-phase generator is given in Tables 1.1 and 1.2, respectively. Eigenvalue was evaluated by considering the same flux level in both three- and six-phase machine. This was ensured by considering the stator voltage of three-phase machine as twice of six-phase machine [27]. Hence, value of the terminal voltage for three-phase and six-phase machine was taken as 240 and 120 V, respectively. Results are given for both machine considering the same load at 50% of rated value, at power factor 0.85 (lagging). It is worthwhile to mention here that it is a difficult to establish a correlation of eigenvalue with machine parameter [15]. It has been considered by changing a machine parameter, keeping other at nominal value and noting the variation in eigenvalue [18].

Table 1.1 Eigenvalues of six-phase synchronous generator.

Nomenclature	Eigenvalues
Stator eigenvalue I	−107.8 ± j104.7
Stator eigenvalue II	−19.2 ± j110.3
Rotor eigenvalue	−5.1 ± j38.2
Real eigenvalue	−9136.3, −703.5, −21.0

Table 1.2 Eigenvalues of three-phase synchronous generator.

Nomenclature	Eigenvalues
Stator eigenvalue	−38.3 ± j103.2
Rotor eigenvalue	−27.9 ± j50.4
Real eigenvalue	−8719.7, −503.3, −10.7

In Equation (1.53), derivative component (i.e., with its elements with subscript p) is indicated by coefficient matrix E, with remaining terms (i.e., subscript k) of linearized machine equations are shown by the coefficient matrix F. Matrices E and F elements are defined in the Appendix.