Poles and zeros meaning: In the s-domain, what does a pole represent, and what does a left-half-plane pole indicate about time-domain behavior?

• A. A pole represents a zero-crossing; left-half-plane indicates a growing response.
• B. A pole represents natural modes; left-half-plane indicates a stable, decaying response.
• C. A pole represents a frequency where gain is zero; left-half-plane indicates instability.
• D. A pole represents system gain at DC; left-half-plane indicates oscillatory behavior.

Answer: (B)

The main idea is that poles in the s-domain correspond to the natural modes of how a system can respond in time. Each pole marks a basic exponential mode in the time-domain response, and its location in the complex plane tells you whether that mode grows, decays, or sustains itself.

If a pole lies in the left half of the s-plane (its real part is negative), the corresponding time-domain term decays like e^(−αt). This means the overall response settles down over time, i.e., the system is stable. If the pole has an imaginary part as well, you get damped oscillations, but the key is that the left-half-plane location guarantees the amplitudes shrink with time.

So a pole represents a natural mode, and a left-half-plane pole indicates a stable, decaying response. This is not about zero-crossings, DC gain, or purely oscillatory behavior—the stability comes from the negative real part, which makes the time-domain terms decay rather than grow.