Overview
Week 12 at a Glance
Time constant (first-order)Natural frequency ωnDamping ratio ζPercent overshootSettling timeRouth-Hurwitz stability
Why it matters in practice. Step response testing is how engineers identify unknown systems and validate controller designs. The ability to look at a measured step response and immediately extract ζ and ωn is a fundamental practical skill.
Objectives
What You Will Be Able to Do
Course objectives (CO) define program-level skills. Module objectives (MO) define specific weekly targets that build toward them.
Course Objectives (CO)
CO7: Derive transfer functions; characterize first- and second-order systems from step response and Bode plot data.
CO8: Assess closed-loop stability using gain and phase margins.
Module Objectives (MO) — Week 12
Determine the time constant of a first-order system from a measured step response and relate it to the system pole location.
CO7
CO7
Compute ωn, ζ, percent overshoot, settling time, and rise time for a second-order system from its transfer function.
CO7
CO7
Classify system damping (overdamped, critically damped, underdamped) from pole locations in the s-plane.
CO7
CO7
Apply the Routh-Hurwitz criterion to determine whether all closed-loop poles lie in the left-half s-plane.
CO8
CO8
Review these objectives before you start each assignment. They map directly to what is assessed on the quiz, homework, and exams.
Suggested Learning Path
How to Work Through This Week
Follow this sequence. Each step prepares you for the next. Do not attempt graded work before completing the instructional material it depends on.
1
Read Alciatore Ch. 8
Work through the second-order step response derivation completely before lecture. The relationships between ζ, ωn, overshoot, and settling time are dense — write them out on an index card to reference.
2
Attend Lecture
Lecture 1 covers first-order response and the time constant. Lecture 2 covers second-order response, performance metrics, and Routh-Hurwitz. Problem sessions include extracting parameters from a plotted step response.
3
Lab: Step Response Identification on Arduino
Apply a step input to a first-order RC circuit and record the step response on your Arduino ADC. Extract the time constant and compare to the calculated value from R and C.
Instructional Material
Required Readings, Videos, and Resources
Complete all required items before moving to graded activities. The Aligns to column maps each resource to the module objectives it directly supports.
| Resource | What You Will Gain | Aligns to | Est. Time |
|---|---|---|---|
| Read Alciatore Ch. 8 — System Response (5th Ed.) |
First-order step response and time constant, second-order standard form, performance metric formulas, pole-zero map, Routh-Hurwitz array construction. | MO1-MO4 | 75 min |
| Lab Lab: First-Order Step Response Identification |
Record step response on Arduino ADC. Extract time constant graphically and compare to theory. | MO1 | ~2 hr lab |
| Watch Micro-lecture: Reading a Step Response in 5 Minutes |
Visual guide to extracting ζ, ωn, overshoot, and settling time from a plotted step response. | MO2 | 5 min |
Graded Learning Activities
Assignments and Due Dates
All graded work is submitted through Canvas. Complete the listed prerequisites before attempting each assignment.
| Assignment | Prerequisites | What Is Assessed | Aligns to | Points |
|---|---|---|---|---|
| Module 5 Homework: Dynamic Systems End of Week 13 |
Complete Ch. 8 reading and attend Week 12 lectures before attempting step response and Routh-Hurwitz problems. | First/second-order TF analysis, step response performance metrics, pole classification, and Routh-Hurwitz stability. | MO1-MO4 | 50 pts |
Academic integrity. Your step response must be recorded from your own RC circuit during the lab session. The time constant you report must be extracted from your measured data, not computed solely from nominal component values.