What does a pole actually do?
Poles are often introduced as roots of a denominator. A better intuition is that a pole is a recipe for how a system naturally evolves in time.
A normalized natural mode, not a complete transfer-function step response. Growing values are clipped at magnitude eight; use the numerical pole location to judge stability.
The horizontal position decides whether the response dies or explodes.
A pole at s = σ contributes an exponential term eσt. If σ is negative, the mode decays. If σ is positive, it grows.
The imaginary part sets oscillation frequency.
A complex pair appears as σ ± jωd. The real part still controls the envelope, while the imaginary part introduces sinusoidal motion inside it.
A pole plot compresses an entire transient into a point.
Fast decay and short-lived transient.
Slow decay and long settling.
Fast oscillation.
Growing mode and instability.
An oscillation needs a conjugate pair.
For a system with real coefficients, complex poles appear in conjugate pairs. Switch to “Real pole” above and the oscillation disappears, leaving pure exponential behavior.