By the end of this section, you should be able to

1. Convert second-order transfer functions into standard form and identify natural frequency and damping ratio

2. Predict overshoot, rise time, and settling time from damping ratio and natural frequency

3. Classify responses as underdamped, critically damped, or overdamped from pole locations and time response

4. Sketch and interpret second-order step responses and relate features to pole locations

5. Validate predictions using simulation and adjust parameters to meet a target response 

6. Modify sketch when "extra" poles or zeros are added

This section builds your core intuition for second-order systems, which show up everywhere in engineering dynamics, vibration, and instrumentation. You will learn how to rewrite a second-order transfer function in standard form, identify the defining parameters that control response, and classify stability from either transfer function form or state-space form by looking at pole locations. From there, the focus shifts to step response sketching. You will practice drawing a clean first-pass sketch, then refining it as you incorporate the key time-domain performance measures used in analysis and design, including 10 percent to 20 percent rise time, peak time, maximum overshoot, and 1 percent settling time. You will also learn how to adjust your sketch and your expectations when the model is no longer a pure second-order system, for example when extra poles or zeros are added. The section includes worked examples that move from qualitative reasoning to more accurate predictions, plus a dedicated connection between state-space representation and the familiar second-order response shape. Use the table of contents below to jump between the introduction, examples, state-space analysis, the extra poles and zeros material, and the lecture code so you can practice the workflow repeatedly until it is automatic.