Mapping constant damping ratio line to z-plane

In the s-plane, a constant damping ratio line is a radial line from the origin making angle $\theta = \cos^{-1}\zeta$ with the negative real axis. On it, $s = -\zeta\omega_n + j\omega_n\sqrt{1-\zeta^2}$, i.e. $s = -\omega_d\dfrac{\zeta}{\sqrt{1-\zeta^2}} + j\omega_d$.

Mapping with $z = e^{Ts}$

Use $T = 2\pi/\omega_s$:

$$z = e^{Ts} = \exp\left(-\frac{2\pi\zeta}{\sqrt{1-\zeta^2}}\cdot\frac{\omega_d}{\omega_s}\right)\angle\ 2\pi\frac{\omega_d}{\omega_s}$$

• $|z| = \exp\left(-\dfrac{2\pi\zeta}{\sqrt{1-\zeta^2}}\cdot\dfrac{\omega_d}{\omega_s}\right)$ decreases exponentially as $\omega_d$ increases.
• $\angle z = 2\pi\omega_d/\omega_s$ increases linearly with $\omega_d$.

Since the radius shrinks exponentially as the angle grows, a constant-ζ line maps into a logarithmic spiral. It starts at $z = 1$ (for $\omega_d = 0$) and winds toward the origin.

Example: $\zeta = 0.5$

At $\omega_d = \omega_s/2$ the spiral reaches the negative real axis. The part from $\omega_d = 0$ to $\omega_s/2$ fills the upper half of the z-plane. The line for $-\omega_d$ maps to the mirror spiral in the lower half.

                Im z
                 |
                 * B = j0.404  (wd/ws = 1/4)
                 |
                 |     * A = 0.449 + j0.449  (1/8)
                 |
 ------*---------+-----------* O ------ Re z
  C = -0.163     0           1   (wd = 0)
  (wd/ws = 1/2)

 zeta = 0.5 spiral runs O -> A -> B -> C inside
 the unit circle; its mirror image (for -wd) lies
 in the lower half-plane.

Key points

• $\zeta = 0$ (the $j\omega$ axis) maps onto the unit circle.
• $\zeta = 1$ (the negative real axis) maps onto the segment $0 < z \le 1$ of the positive real axis.
• Larger $\zeta$ gives a spiral that shrinks faster, i.e. tighter toward the origin.
• In design, the desired closed-loop poles lie on the ζ-spiral at angle $\omega_dT = 2\pi/N$, where $N$ is the number of samples per cycle of damped oscillation.