s-plane

The s-plane is the complex plane in which the Laplace transform’s variable $s=\sigma +j\omega$ lives. Real part $\sigma$ on the horizontal axis, imaginary part $\omega$ (or $j\omega$) on the vertical axis.

The s-plane is the geometric setting for everything to do with Laplace-domain analysis: poles and zeros, regions of convergence, stability, frequency response.

The vertical axis is the Fourier axis

On the imaginary axis $\sigma =0$, we have $s=j\omega$ — purely imaginary frequencies. The Laplace transform restricted to this axis is the Fourier transform (in ω-form):

$X(j\omega )=X(s){|}_{s=j\omega }.$

So the Fourier transform is the “shadow” of the Laplace transform on the vertical axis of the s-plane. The full Laplace transform lives in the two-dimensional plane and gives the Fourier transform when you slice it along the imaginary axis.

For a stable system, the imaginary axis is inside the ROC of $H(s)$, and $H(j\omega )$ is well-defined. For an unstable system, the imaginary axis isn’t in the ROC, and the Fourier transform of $h$ doesn’t exist.

Pole locations and signal behavior

The position of a pole in the s-plane tells you the corresponding time-domain signal’s shape:

• Pole at $s=-\alpha$ on the negative real axis ($\alpha >0$): decaying exponential $e^{-\alpha t}u(t)$. The further left, the faster the decay.
• Pole at $s=\alpha$ on the positive real axis ($\alpha >0$): growing exponential $e^{\alpha t}u(t)$.
• Pole at $s=0$: a step (or for a double pole, a ramp).
• Pair of complex-conjugate poles at $s=-\alpha \pm j{\omega }_0$: damped sinusoid $e^{-\alpha t}\cos ({\omega }_{0}t+\varphi )u(t)$. Real part determines decay rate; imaginary part determines oscillation frequency.
• Pair of conjugate poles on the imaginary axis ($\alpha =0$): undamped sinusoid $\cos ({\omega }_{0}t+\varphi )$.
• Pair of conjugate poles in the right half-plane ($\alpha <0$): growing oscillation — instability.

This geometric language is one of the most useful pictures in the course. Engineers learn to read a system’s behavior from its pole-zero plot at a glance.

Stability regions

• Open left half-plane ($\sigma <0$): poles here mean decaying or oscillating-and-decaying transients. Stable.
• Imaginary axis ($\sigma =0$): poles here give sustained oscillations. Marginally stable (still bounded, but doesn’t return to zero).
• Right half-plane ($\sigma >0$): poles here give growing transients. Unstable.

The standard stability requirement for a causal LTI system is all poles strictly in the open left half-plane. Equivalent statements: $\int |h|dt<\infty$, the system is BIBO stable, the ROC includes the imaginary axis.

Damping ratio and natural frequency

For a complex-conjugate pole pair at $-\alpha \pm j{\omega }_d$ (with $\alpha =\zeta {\omega }_n$ for some damping ratio $\zeta$ and natural frequency ${\omega }_n$), the geometry on the s-plane encodes the dynamics:

• Distance from the origin: natural frequency ${\omega }_n$.
• Angle from the negative-real axis: ${\cos }^{-1}\zeta$, the damping angle. Poles on the negative real axis correspond to critical or overdamped systems; poles near the imaginary axis correspond to lightly-damped, highly oscillatory systems.

This is the language of second-order systems and control theory.