CHEN90032 · Process Simulation and Control
Frequency Response, Bode Plots & Stability Margins
This chapter of the University of Melbourne CHEN90032 Process Simulation and Control guide covers how a stable process treats a sustained sinusoid: the output is a sinusoid of the same frequency, scaled by the amplitude ratio AR = |G(jω)| and shifted by the phase φ = ∠G(jω). You will learn to build AR and φ for first-order, second-order, dead-time and PID elements, apply the Bode stability criterion (φ = −180° at the crossover frequency ω_c), and compute the ultimate gain and period plus the gain and phase margins that decide how robust a control loop is. This is the frequency-domain half of the exam's transfer-function question and the input to the controller-tuning chapter that follows.
What this chapter covers
- 01Substitute s → jω and read AR = |G(jω)| and φ = ∠G(jω) from a transfer function
- 02Write AR and φ for a first-order lag K/(τs+1), including the break frequency ω_b = 1/τ where φ = −45°
- 03Write AR and φ for a second-order element and for a pure time delay e^(−θs) (AR = 1, φ = −ωθ in radians)
- 04Combine blocks in series: multiply the amplitude ratios and add the phases
- 05Give the frequency response of P, PI, PD and PID controllers and see how integral lags and derivative leads the phase
- 06Apply the Bode stability criterion: a loop is stable if AR_OL(ω_c) < 1 at the frequency where φ_OL = −180°
- 07Compute the ultimate gain K_cu = 1/AR_p(ω_c) and ultimate period P_u = 2π/ω_c for P-only control
- 08Compute the gain margin GM = 1/AR_OL(ω_c) and phase margin PM = 180° + φ_OL(ω_g)
- 09Find the maximum extra dead time Δθ_max = (PM/ω_g)(π/180°) a loop can absorb before instability
- 10Recognise why a pure first- or second-order loop has no finite ultimate gain, but a delay or third lag creates one
Gain margin, ultimate gain and ultimate period of a three-lag loop
- +1Phase-crossover ω_c: three equal lags give φ_OL = −3 tan⁻¹(ω). Set φ_OL = −180°, so tan⁻¹(ω_c) = 60° and ω_c = tan 60° = √3 = 1.73 rad/s.
- +1Amplitude ratio of one lag at ω_c: AR = 1/√(ω_c²+1) = 1/√(3+1) = 1/2 = 0.5.
- +1Open-loop AR at ω_c: multiply the three lags and the gain, AR_OL(ω_c) = K_c × (0.5)³ = 2 × 0.125 = 0.25.
- +1Gain margin GM = 1/AR_OL(ω_c) = 1/0.25 = 4.0 (dimensionless). Since GM > 1 the loop is stable, and 4.0 sits at the top of the healthy 1.7–4.0 range.
- +1Ultimate gain: at the boundary AR_OL(ω_c) = 1, so K_cu × 0.125 = 1, giving K_cu = 8. This is a cross-check: K_cu = K_c × GM = 2 × 4.0 = 8.
- +1Ultimate period P_u = 2π/ω_c = 6.283/1.732 = 3.63 s.
Key terms
- Amplitude ratio (AR)
- The magnitude |G(jω)| of the frequency response: the ratio of output amplitude to input amplitude for a sustained sinusoid at frequency ω. It carries the units of the process gain (output units per input units).
- Phase angle (φ)
- The argument ∠G(jω) of the frequency response: the phase shift between input and output sinusoids. A negative φ is a phase lag (the output peaks after the input). Quoted in degrees on a Bode plot, but kept in radians inside a delay term e^(−jωθ).
- Phase-crossover frequency (ω_c)
- The frequency at which the open-loop phase reaches −180°. The Bode stability criterion is evaluated here: the closed loop is stable if and only if AR_OL(ω_c) < 1.
- Gain-crossover frequency (ω_g)
- The frequency at which the open-loop amplitude ratio equals 1. The phase margin is read here. For a stable loop ω_g is below ω_c.
- Ultimate gain and period (K_cu, P_u)
- Under P-only control, K_cu = 1/AR_p(ω_c) is the controller gain that just drives the loop to a sustained oscillation, and P_u = 2π/ω_c is that oscillation's period. Both feed the Ziegler–Nichols and Tyreus–Luyben tuning rules.
- Gain margin (GM)
- GM = 1/AR_OL(ω_c): the dimensionless factor by which the loop gain can be multiplied before instability. GM > 1 is stable, = 1 is marginal, < 1 is unstable; with no controller GM equals K_cu.
- Phase margin (PM)
- PM = 180° + φ_OL(ω_g): how far the phase sits above −180° at the gain-crossover. A positive PM means stability with room to spare; a healthy target is roughly 30°–45°.
- Maximum additional dead time (Δθ_max)
- Δθ_max = (PM/ω_g)(π/180°): the largest extra transport delay the loop can tolerate before going unstable, because a delay adds phase lag −ωθ that erodes the phase margin. PM in degrees and ω_g in rad/time give a result in time units.
Frequency Response, Bode Plots & Stability Margins FAQ
Why is stability decided at the frequency where the phase is −180°?
At the phase-crossover frequency ω_c the feedback signal returns exactly inverted, so it reinforces a fresh disturbance instead of cancelling it. If the loop also has enough gain there (AR_OL(ω_c) ≥ 1) the oscillation sustains or grows and the closed loop is unstable. If AR_OL(ω_c) < 1 the returning signal is smaller each cycle and the loop is stable — that is the whole Bode criterion.
What is the difference between the gain margin and the phase margin?
They measure the same distance-to-instability at two different frequencies. The gain margin GM = 1/AR_OL(ω_c) is read where the phase is −180° and says how much the loop gain can rise before instability. The phase margin PM = 180° + φ_OL(ω_g) is read where AR = 1 and says how much extra phase lag (for example from added dead time) the loop can absorb. A robust design keeps both positive — typically GM around 1.7–4.0 and PM around 30°–45°.
Can AI help me with frequency response and stability margins in CHEN90032?
Yes, as a study aid rather than an answer service. Sia can explain, step by step, how to substitute s → jω, build AR and φ for each block, solve φ_OL = −180° for ω_c, and compute the gain and phase margins, and it can check your reasoning against your own worked attempt. It will not sit your closed-book exam for you, and no tool can promise a specific grade; the goal is to help you understand the method so you can reproduce it under exam conditions.
Studying with AI? Sia — free AI chemical engineering tutor works through CHEN90032 step by step.
Exam move
Drill one routine until it is automatic: assemble the open-loop transfer function, substitute s → jω, then multiply the amplitude ratios and add the phases. Everything else follows. Solve φ_OL = −180° for the phase-crossover ω_c (iterate in radians when a dead time is present), which gives the gain margin GM = 1/AR_OL(ω_c) and, for P-only control, the ultimate gain K_cu = 1/AR_p(ω_c) and period P_u = 2π/ω_c. Then solve AR_OL = 1 for the gain-crossover ω_g to get the phase margin PM = 180° + φ_OL(ω_g) and the spare dead time Δθ_max = (PM/ω_g)(π/180°). Guard the recurring slips: a delay's phase is in radians, the gain margin lives at ω_c while the phase margin lives at ω_g, and only a delay or a third lag gives a finite ultimate gain. The written final is closed-book with a provided formula sheet (the Δθ_max relation and the tuning rules are on it), so practise the method and carry units and the sign of φ throughout — that is where the marks are. Pace at about 1.8 minutes per mark (3 hours over 100 marks). Confirm the exam date for the ~June 2027 end-of-Semester-1 exam period on Canvas.
