Where it shows up
This one shape recurs wherever a system stores energy in two places and trades it back and forth. A mass on a spring, an RLC circuit swapping charge between capacitor and inductor, a car suspension, a drone holding altitude, the recovery of a control loop after a knock: each collapses to a pole pair and a damping ratio. The zero appears whenever a fast path bypasses the slow dynamics, as in a PD controller or a lead compensator, where it lifts the high-frequency gain and drags the transient a little earlier in time.
The knobs
- Natural freq ωn: how far the poles sit from the origin. Higher values speed the whole response up and slide the Bode peak to the right.
- Damping ζ: slides the poles left or right. Below zero is unstable, near zero rings hard, and at 1 the pair goes real and critically damped.
- Add real zero: switches on the amber zero, adding a numerator factor that lifts the high-frequency gain and sharpens the early transient.
- Zero location: where the zero sits on the negative real axis. The closer it creeps to the origin, the stronger its pull on the response.
- Show Bode magnitude: toggles the lower frequency-response panel so the step response can stretch to fill the space.