This article cannot start without a short exposé of the s‑plane. It is a map of all possible complex frequencies s = σ + jω. The horizontal axis (σ) tells you whether a signal grows or decays over time. The vertical axis (jω) tells you how fast it oscillates — its frequency.
A pole, marked with ×, is a point where the system naturally resonates. If the pole lies in the left half‑plane (σ negative), that resonance dies out on its own — a stable system. If the pole lies in the right half‑plane (σ positive), the resonance grows without bound — unstable. If the pole sits exactly on the imaginary axis (σ = 0), the system oscillates forever at a constant amplitude, like an ideal oscillator.
Because real systems have real coefficients, complex poles always appear in conjugate pairs — one above the real axis, one below. That's why one sees them as symmetric pairs, like the green ×s in the figure. Real poles (those on the horizontal axis) stand alone — like the red × in the figure.
Zeros, marked with ○, are points where the transfer function becomes zero. They do not cause instability. Instead, they shape the frequency response — cutting off certain frequencies or boosting others. If poles are the engine of the system, zeros are the sculptor.
Understanding the s‑plane starts with an observation: left means stable, right means unstable. The rest — damping, natural frequency, response shape — follows directly from where the poles sit on this map.