Control systems — Nichols chart interpretation In classical frequency-response design, the Nichols chart plots what relationship to help map open-loop data to closed-loop performance?

Introduction / Context:
The Nichols chart is a cornerstone tool in classical control engineering. It converts frequency-domain information (open-loop gain and phase) into direct insight about closed-loop behavior using families of constant closed-loop magnitude (M-circles) and constant closed-loop phase (N-circles). Designers can judge stability margins and performance straight from the open-loop data.

Given Data / Assumptions:

• We are dealing with linear time-invariant (LTI) feedback systems.
• Open-loop frequency response is plotted as log magnitude (dB) versus phase angle.
• Nichols chart overlays closed-loop contours on this open-loop plane.

Concept / Approach:
The chart maps the point L(jω) = G(jω)H(jω) (open-loop) into the closed-loop transfer T(jω) = L(jω)/(1 + L(jω)). M- and N-circles show how any open-loop point translates into closed-loop magnitude and phase. Thus the chart inherently relates closed-loop values to open-loop data, enabling rapid loop-shaping and robustness assessment.

Step-by-Step Solution:
Recognize that Nichols axes are open-loop gain (dB) versus open-loop phase (degrees).Identify that the superposed circles correspond to constant closed-loop responses.Therefore the chart relates closed-loop performance to open-loop frequency response.

Verification / Alternative check:
From T = L/(1 + L), loci of constant |T| and ∠T in the L-plane generate the M and N contours drawn on the Nichols chart.

Why Other Options Are Wrong:

• (a) and (b): Restrict to first- or second-order only; Nichols is general.
• (d): Not limited to controller curves; it is a system-level mapping tool.

Common Pitfalls:

• Confusing Nichols with Bode or Nyquist; Nichols uniquely uses dB vs phase with closed-loop overlays.

Final Answer:
Closed-loop performance contours overlaid on open-loop gain (in dB) versus phase