First-order plus dead time (FOPTD) system: a unity-gain process has time constant τ = 5 min. If the phase lag at excitation frequency ω = 0.2 rad/min is 60°, what is the dead time θ (minutes)?

Introduction / Context:
Estimating the dead time of a first-order plus time delay (FOPTD) model from frequency-response data is a practical control task. Phase measurements from sine tests are particularly informative because dead time adds a linear-in-frequency phase lag to the system’s inherent lag.

Given Data / Assumptions:

• Process model: G(jω) = exp(−jωθ) / (1 + jωτ), unity gain.
• Given τ = 5 min, ω = 0.2 rad/min.
• Measured phase lag |φ| = 60° = π/3 radians.

Concept / Approach:
For a FOPTD element, the phase lag is φ(ω) = −[arctan(ωτ) + ωθ]. The first term is the first-order lag; the second term arises from the dead time. Knowing φ and τ at a particular ω allows solving for θ directly.

Step-by-Step Solution:
Compute arctan(ωτ) with ωτ = 0.2 * 5 = 1 ⇒ arctan(1) = π/4.Write phase equation magnitude: π/3 = π/4 + ωθ ⇒ ωθ = π/3 − π/4 = (4π − 3π)/12 = π/12.Solve for θ: θ = (π/12)/ω = (π/12)/0.2 = (π/12) * 5 = 5π/12 minutes.Thus, θ = 5π/12 ≈ 1.309 minutes.

Verification / Alternative check:
Plug back into φ = −[arctan(1) + 0.2 * (5π/12)] = −[π/4 + π/12] = −π/3, which matches the measured −60° lag.

Why Other Options Are Wrong:
π/6 or π/12: These are intermediate angles, not the dead time in minutes; they arise before dividing by ω = 0.2.π/3: This is the total phase in radians, not θ.π/8: Does not satisfy the phase equation when substituted.

Common Pitfalls:
Forgetting to convert degrees to radians; failing to divide the residual phase by ω; or ignoring that the process time constant contributes arctan(ωτ) to the phase independently of dead time.

Final Answer:
5π/12