Transient Response on the Z-Plane
In order to draw equivalent conclusions on the z-plane we now map those lines through z = esT.
The vertical lines on the s-plane are lines of constant settling time and are characterized by the equation s = σ1 jω, where the real part, σ1 = —4/Ts, is constant and is in the left-half-plane for stability. Substituting this into z = esT, we obtain
Equation (1) denotes concentric circles of radius r1. If σ1 is positive, the circle has a larger radius than the unit circle. On the other hand, if σ1is negative, the circle has a smaller radius than the unit circle. The circles of constant settling time, normalized to the sampling interval, are shown in Figure 1 with radius = e-4/(Ts/T) . Also, Ts/T = -4/ln(r), where r is the radius of the circle of constant settling time. The horizontal lines are lines of constant peak time. The lines are characterized by the equation s = σ1 jω, where the imaginary part, ω1 =π/TP, is constant. Substituting this into z = esT, we obtain
Equation (2) represents radial lines at an angle of θ1.if σ is negative, that section of the radial line lies inside the unit circle. If σ is positive, that section of the radial line lies outside the unit circle. The lines of constant peak time normalized to the sampling interval are shown in Figure 1. The angle of each radial line is
Finally, we map the radial lines of the s-plane onto the z-plane. Remember, these radial lines are lines of constant percent overshoot on the s-plane. From Figure 2, these radial lines are represented by
(3) , (4) And (5)
Transforming Eq. (4) to the z-plane yields
Thus, given a desired damping ratio ζ, Eq. (5) can be plotted on the z-plane through a range of ωT as shown in Figure 1. These curves can then be used as constant percent overshoot curves on the z-plane.
FIGURE 1 Constant damping ratio, normalized settling time, and normalized peak time plots on the z-plane
FIGURE 2 the s-plane sketch of constant percent overshoot line