Corresponding Z-transform:

$\displaystyle z$	$\displaystyle =$	$\displaystyle e^{Ts}$
	$\displaystyle =$	$\displaystyle e^{jwT}$

When = constant, it represents a straight line from the origin at an angle of $\theta = wT$ rad, measured from positive real axis as shown in Figure 3 (b).

Constant damping ratio loci: If $\xi$ denotes the damping ratio:

$\displaystyle s$	$\displaystyle =$	$\displaystyle -\xi w_{n} \pm jw_{n} \sqrt{1-\xi^{2}}$
	$\displaystyle =$	$\displaystyle -\frac{w}{\sqrt{1-\xi^{2}}} \xi \pm j w$
	$\displaystyle =$	$\displaystyle - w \tan\beta \pm j w$

is the natural undamped frequency and $\beta=\sin^{-1} \xi$ . If we take Z-transform

$\displaystyle z$	$\displaystyle =$	$\displaystyle e^{T(-wtan \beta + jw)}$
	$\displaystyle =$	$\displaystyle e^{-2\pi w \tan \beta /w_s} \angle{(2\pi w/w_s)}$

For a given $\xi$ or $\beta$ , the locus in s-plane is shown in Figure 4(a).

In z-plane, the corresponding locus will be a logarithmic spiral as shown in Figure 4(b), except for $\xi=0$ or $\beta=0^o$ and $\xi=1$ or $\beta=90^o$ .

$\begin{figure}\centering \input{m2l2f4.pstex_t}\end{figure}$
Figure 4: Constant damping ratio locus in (a) s-plane and (b) z-plane