DAMPING RATIO AND UNDAMPED NATURAL FREQUENCY IN THE z-PLANE
Damping Ratio
As shown in Figure 7.9(a), lines of constant damping ratio in the s-plane are lines where ζ = cos α for a given damping ratio. The locus in the z-plane can then be obtained by the substitution z = esT . Remembering that we are working in the third and fourth quadrants in
Undamped Natural Frequency
As shown in Figure 7.11, the locus of constant undamped natural frequency in the s-plane is a circle with radius ωn . From this figure, we can write
The locus of constant ωn in the z-plane is given by (7.7) and is shown in Figure 7.10 as the vertical lines. Notice that the curves are given for values of ωn ranging from ωn = π/10T to ωn = π/ T .
Notice that the loci of constant damping ratio and the loci of undamped natural frequency are usually shown on the same graph.