Evaluate the statement: 'The phase margin of a control system and the damping ratio of its transient response have no relationship.' Indicate whether this statement is true or false.

Introduction / Context:
Phase margin and damping ratio are fundamental stability measures in control systems. Phase margin is a frequency-domain specification obtained from Bode or Nyquist plots, while damping ratio is a time-domain parameter that affects overshoot and settling time. There exists a practical relationship between the two, particularly for second-order dominant systems, where phase margin can be related to damping ratio using approximate formulas.

Given Data / Assumptions:

• We are analyzing a unity feedback system with dominant complex-conjugate poles.
• Phase margin (PM) is defined as 180° + phase(G(jω_gc)) at the gain crossover frequency.
• Damping ratio (ζ) is defined from the standard second-order system transfer function response.

Concept / Approach:
Approximate correlation exists: PM ≈ tan⁻¹(2ζ / √(√(1+4ζ⁴) − 2ζ²)). This relation shows that higher damping ratios (ζ ~ 0.7) correspond to larger phase margins (~60°), yielding well-damped, low-overshoot responses. Conversely, very low damping ratios produce small phase margins and oscillatory responses.

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