Nichols Chart

The Nichols chart is a graphical tool in control system design that displays the relationship between open-loop gain (in dB) and open-loop phase (in degrees) on a single plane. It allows the designer to read closed-loop magnitude and phase characteristics directly from the open-loop data, making it a compact and powerful alternative to separate Bode and Nyquist plots for closed-loop performance assessment.

Core Concept of the Nichols Chart

The Nichols chart is a transformed version of the complex plane where the horizontal axis represents the open-loop phase angle in degrees and the vertical axis represents the open-loop gain in decibels. A system's frequency response is plotted as a curve on this plane, with frequency w as a parameter along the curve. The critical point in the Nyquist plane (-1+j0) corresponds to the point (-180 degrees, 0 dB) in the Nichols chart.

Overlaid on this plane are two families of contours: M-circles (constant closed-loop magnitude contours) and N-circles (constant closed-loop phase angle contours). These contours are derived from the relationship between open-loop and closed-loop transfer functions. When the open-loop locus is plotted on the Nichols chart, its intersection with these contours immediately reveals the closed-loop frequency response without any additional computation.

The M and N contours in the Nichols chart correspond to the M-circles and N-circles in the complex Nyquist plane, but mapped onto the gain-phase coordinate system. This transformation makes it easier to read off gain margin (the vertical distance from the open-loop curve's -180 degree crossing to 0 dB) and phase margin (the horizontal distance from the 0 dB crossing to -180 degrees).

M-Circles and N-Circles: Mathematical Basis

For a unity feedback system with open-loop transfer function G(jw), the closed-loop transfer function is T(jw) = G(jw) / (1 + G(jw)). If we write G(jw) = x + jy in rectangular form, the locus of constant closed-loop magnitude M = |T(jw)| in the x-y plane forms circles called M-circles. The center of the M-circle is at (-(M squared)/(M squared - 1), 0) and its radius is M/(|M squared - 1|). When M = 1 (0 dB closed-loop), the M-circle degenerates to the line x = -0.5.

Similarly, the locus of constant closed-loop phase angle N = tan(angle T(jw)) forms circles called N-circles centered at (-0.5, 1/(2N)). Both families of circles are centered on specific locations relative to the -1+j0 critical point, confirming that the critical point is the origin of all stability information. In the Nichols chart representation, these circular loci in the Nyquist plane become the distorted oval contours seen on the standard Nichols chart grid.

Reading Closed-Loop Performance from the Nichols Chart

Once the open-loop Nichols locus is plotted, closed-loop performance is read by noting which M-contour is tangent to the locus. The tangent M-contour gives the resonance peak Mr, which is the maximum closed-loop magnitude. The frequency at which tangency occurs is the resonant frequency wr. Systems with Mr greater than 1.3 (about 2.3 dB) often exhibit unacceptable overshoot in transient response.

The bandwidth of the closed-loop system is found by identifying where the locus crosses the -3 dB M-contour. This is directly readable from the Nichols chart without any additional computation, making it highly efficient for design iterations where gain is adjusted by simply shifting the locus up or down on the chart.

Solved Numerical Example

Given an open-loop system G(s) = 10 / (s(s+1)(s+2)), find the gain and phase at w = 1 rad/s for plotting on the Nichols chart, and determine the gain margin.

Given:
G(s) = 10 / [s(s+1)(s+2)]
Frequency of interest: w = 1 rad/s, w_pc to find GM

Why this formula applies:
Nichols chart plots |G(jw)| in dB vs angle G(jw) in degrees.
GM is the gain reduction required to make |G| = 0 dB at phase = -180 deg.

Formula:
Phase crossover: angle G(jw_pc) = -180 deg
-90 - arctan(w) - arctan(w/2) = -180
arctan(w) + arctan(w/2) = 90

Substitution:
Using: (w + w/2)/(1 - w^2/2) = tan(90) => infinity
=> 1 - w^2/2 = 0 => w_pc = √2 rad/s

Calculation at w_pc = √2:
|G(j√2)| = 10 / [√2 * √(2+1) * √(2+4)]
= 10 / [√2 * √3 * √6]
= 10 / [√36] = 10/6 = 1.667

Gain at phase crossover = 20 log(1.667) = 4.44 dB

Gain Margin = 0 - 4.44 = -4.44 dB (UNSTABLE for K=10)

For stability: K < 10/1.667 = 6 (approx.)

Final Answer: GM = -4.44 dB; system with K=10 is unstable.
On Nichols chart: locus crosses -180 deg line at +4.44 dB, above 0 dB => unstable.

Exam Tip: On the Nichols chart, if the open-loop locus passes to the right of the critical point (-180 deg, 0 dB), the system is stable for a minimum-phase open-loop system. Gain margin is the vertical distance from the -180 degree axis crossing to 0 dB; if the crossing is above 0 dB, gain margin is negative and the system is unstable.

Mechanism Summary

• The Nichols chart maps open-loop data (gain in dB, phase in degrees) onto a single plane with pre-drawn closed-loop M and N contours.
• The critical point -1+j0 of the Nyquist plane corresponds to the point (-180 degrees, 0 dB) in the Nichols chart.
• Gain margin is read as the vertical distance from the open-loop locus crossing at -180 degrees to the 0 dB line. Positive GM means the crossing is below 0 dB.
• Phase margin is read as the horizontal distance from the open-loop locus crossing at 0 dB to the -180 degree line.
• Changing the gain K shifts the entire locus vertically on the Nichols chart without changing its shape, making gain design very efficient.
• The M-contour tangent to the locus gives the peak closed-loop magnitude Mr. Crossing the -3 dB contour gives the closed-loop bandwidth directly.

Quick Revision

• Nichols chart plots open-loop gain (dB) vs open-loop phase (degrees) with overlaid closed-loop M and N contours.
• Critical point -1+j0 equivalent is (-180 deg, 0 dB) on the Nichols chart.
• GM = distance below 0 dB at phase = -180 deg crossing. PM = distance right of -180 deg at gain = 0 dB crossing.
• Gain changes shift the locus vertically only. No reshaping is needed, making iterative gain design fast.
• Resonance peak Mr = highest M-contour tangent to the locus. Bandwidth = where locus crosses -3 dB M-contour.
• Common trap: confusing the Nichols chart with the Bode plot. Bode plots magnitude and phase separately; Nichols plots them against each other.
• M-circles in Nyquist plane become distorted ovals in Nichols plane due to the dB and degree transformation.