CHEN90032 · Process Simulation and Control
Second-Order & Complex Process Dynamics
Second-order dynamics is the part of CHEN90032 Process Simulation and Control at the University of Melbourne where a process gains a second energy or mass store and can start to overshoot and oscillate. You model it with the standard transfer function K/(τ²s² + 2ζτs + 1), read the damping coefficient ζ to classify the response as over-, critically- or under-damped, and — when it rings — pull the overshoot, decay ratio, peak time and period straight from ζ and τ.
What this chapter covers
- 01Write and interpret the standard second-order transfer function K/(τ²s² + 2ζτs + 1), with correct symbols and units
- 02Read the time constant τ and damping coefficient ζ straight off any second-order denominator
- 03Classify a response from ζ as over-damped (ζ>1), critically damped (ζ=1), under-damped (0<ζ<1) or undamped
- 04Combine two first-order lags in series: τ = √(τ₁τ₂) and ζ = (τ₁+τ₂)/(2√(τ₁τ₂))
- 05Compute the under-damped descriptors: overshoot OS, decay ratio DR, time to first peak tp and period P
- 06Run the descriptors backwards — from a measured overshoot to ζ, and from the period to τ
- 07Fit a second-order model with Smith's method using the 20% and 60% response times (t20/t60)
- 08Avoid the common marked slips: the 2ζτ factor, the overshoot baseline, and time-unit mixing
Two first-order tanks in series → find K, τ, ζ and classify
- +1Write the series transfer function — cascaded blocks multiply: G(s) = K₁K₂ / [(τ₁s+1)(τ₂s+1)] = 3 / [(6s+1)(2s+1)], so K = K₁K₂ = 3 (output units per input unit).
- +1Expand the denominator: (6s+1)(2s+1) = 12s² + 6s + 2s + 1 = 12s² + 8s + 1.
- +1Match the s² term to the standard form τ²s² + 2ζτs + 1: τ² = 12, so τ = √12 = 3.46 min (keep minutes — the data are in minutes).
- +1Match the s term: 2ζτ = 8, so ζ = 8 / (2 × 3.46) = 8 / 6.93 = 1.15 (dimensionless). Cross-check: τ = √(τ₁τ₂) = √12 = 3.46 min and ζ = (τ₁+τ₂)/(2√(τ₁τ₂)) = 8/6.93 = 1.15.
- +1Classify: ζ = 1.15 > 1, so the system is over-damped — two real poles, a smooth approach with no overshoot.
Key terms
- Second-order transfer function
- The model K/(τ²s² + 2ζτs + 1) relating an output to an input for a process with two stores; its behaviour is fixed by the gain K, time constant τ and damping coefficient ζ.
- Damping coefficient ζ
- A dimensionless number that sets the shape of the response. ζ > 1 is over-damped, ζ = 1 critically damped, 0 < ζ < 1 under-damped (oscillatory), ζ = 0 undamped.
- Time constant τ
- Sets the natural speed of the response; it has units of time and equals the square root of the s² coefficient in the standard form. Keep it in the same time units as the data.
- Over-damped / critically damped / under-damped
- The three stable regimes: over-damped (ζ > 1, two real poles, no overshoot), critically damped (ζ = 1, fastest smooth approach), under-damped (0 < ζ < 1, complex poles, overshoots and rings down).
- Overshoot (OS)
- For an under-damped response, the first peak height above the final value divided by the total change, OS = exp(−πζ/√(1−ζ²)). Larger for smaller ζ.
- Decay ratio (DR)
- The ratio of the second overshoot to the first, DR = OS² = exp(−2πζ/√(1−ζ²)); it measures how fast the oscillation dies away.
- Time to first peak (tp) and period (P)
- For an under-damped response, tp = πτ/√(1−ζ²) and the peak-to-peak period P = 2πτ/√(1−ζ²) = 2·tp.
- Smith's method (t20/t60)
- An empirical fit of a second-order model from a step response: the ratio of the times to reach 20% and 60% of the total change gives ζ from a provided chart, and t60 then sets τ.
Second-Order & Complex Process Dynamics FAQ
Is second-order dynamics a big part of the CHEN90032 exam?
It sits inside the transfer-function and frequency-response material, which was a 35-mark question in a representative past paper. The final exam is worth 60% and is a hurdle you must pass to pass the subject: 4 questions / 100 marks, 15 minutes reading plus 3 hours writing, closed-book with a provided formula sheet (Laplace-transform table, tuning rules, Padé approximants, Routh arrays and the Smith's-method chart) and a Casio FX82 calculator. Note that the overshoot, decay-ratio, peak-time and period formulae are NOT on the provided sheet, so memorise them. Confirm the exam date on Canvas — this unit next runs in Semester 1, 2027 (S1 exam period, around June).
How do I get the damping coefficient ζ out of a transfer function?
Put the denominator in the standard form τ²s² + 2ζτs + 1 and match coefficients. The coefficient of s² is τ², so τ = √(that coefficient); the coefficient of s is 2ζτ, so ζ = (coefficient of s)/(2τ). The most common marked slip is dividing by 2 instead of 2τ — always divide by 2τ.
Can AI help me with second-order process dynamics in CHEN90032?
Yes — Sia, the AskSia AI tutor, can explain step by step how to read τ and ζ off a denominator, classify a response, or derive the overshoot and period, and can check your reasoning on a practice problem. Use it to understand the method and build your own working; it does not hand back homework or exam answers, and no tool can promise a particular grade or a pass on a hurdle exam — the marks come from your own shown working.
Studying with AI? Sia — free AI chemical engineering tutor works through CHEN90032 step by step.
Exam move
Anchor everything to one move: put the denominator into the standard form K/(τ²s² + 2ζτs + 1), then read τ = √(s² coefficient) and ζ = (s coefficient)/(2τ). Once ζ is in hand the regime is automatic (over-, critically- or under-damped), and for an under-damped response the four descriptors — tp = πτ/√(1−ζ²), P = 2·tp, OS = exp(−πζ/√(1−ζ²)) and DR = OS² — all follow; drill running them backwards too (measured overshoot → ζ, period → τ), since identification questions are common. Practise Smith's method for responses with no clear peaks, and rehearse the unit discipline that this exam rewards: keep τ, tp and P in the same time units, because the paper mixes minutes and seconds. Budget your time at the exam's own rate of 1.8 minutes per mark (3 hours writing ÷ 100 marks), and write out the standard form, the matched coefficients and the units every time, because the marks are awarded for shown working.
