Ideal PID controller Gc(s) = Kc [ 1 + 1/(τi s) + τd s ] with Kc = 1, τi = 0.5 s, τd = 0.2 s. At what angular frequency ω (rad/s) is the controller's magnitude |Gc(jω)| equal to 1?

Introduction / Context:
For an ideal PID Gc(s) = 1 + 1/(τi s) + τd s (with Kc = 1), the frequency response determines how the controller shapes loop gain and phase. A useful reference point is the frequency where the magnitude of the controller is unity, which helps anticipate closed-loop crossover behavior.

Given Data / Assumptions:

• Kc = 1, τi = 0.5 s, τd = 0.2 s.
• We work with Gc(jω) = 1 + 1/(jω τi) + jω τd.
• We seek |Gc(jω)| = 1.

Concept / Approach:
Write Gc(jω) = 1 + j(ω τd − 1/(ω τi)). The real part is 1; the imaginary part is ω τd − 1/(ω τi). The magnitude |Gc| = sqrt{ 1^2 + [ω τd − 1/(ω τi)]^2 }. Setting |Gc| = 1 forces the imaginary part to zero: ω τd − 1/(ω τi) = 0. Hence ω = 1/√(τi τd). This is the frequency where derivative and integral contributions balance, leaving the real part equal to 1 exactly.

Step-by-Step Solution:
Set imaginary part to zero: ω τd = 1/(ω τi).Solve for ω: ω^2 = 1/(τi τd) ⇒ ω = 1/√(τi τd).Compute with τi = 0.5, τd = 0.2: τi τd = 0.1 ⇒ ω ≈ 3.162 rad/s.

Verification / Alternative check:
At ω = 1/√(τi τd), Gc(jω) = 1 + j(0) ⇒ |Gc| = 1 exactly for Kc = 1; small numerical checks confirm the balance.

Why Other Options Are Wrong:
τi/τd or τd/τi: Wrong functional form and units.1/(τi τd): Misses the square root, overestimates frequency.1/τi: Ignores derivative time constant entirely.

Common Pitfalls:
Forgetting that |a + jb| = 1 implies b = 0 when a = 1; otherwise magnitude exceeds 1. Also, keep units consistent—ω is in rad/s.

Final Answer:
ω = 1/√(τi τd) ≈ 3.16 rad/s