S-Plane

The Complex Plane used in Laplace Transform analysis, where $s=\sigma +j\omega$.

$$s=\underset{\text{damping}}{\underbrace{\sigma }}+j\underset{\text{frequency}}{\underbrace{\omega }}$$

The real axis $\sigma$ controls exponential growth or decay; the imaginary axis $j\omega$ controls oscillation frequency. Setting $\sigma =0$ recovers the Fourier Transform along the imaginary axis.

Stability Regions

Region	Condition	Behavior
Left half-plane	$\sigma <0$	Stable (decaying)
Imaginary axis	$\sigma =0$	Marginally stable (oscillatory)
Right half-plane	$\sigma >0$	Unstable (growing)

The discrete-time analog is the Z-Plane, related by $z=e^{sT}$. The left half of the $s$-plane maps to the interior of the unit circle in the $z$-plane.