Nyquist Plot Construction

The Nyquist plot is a complete frequency response diagram that maps the entire Nyquist contour in the s-plane to the G(s)H(s) plane. Unlike the polar plot which only covers positive frequencies, the Nyquist plot includes contributions from negative frequencies and any semicircular indentations around poles on the imaginary axis. This completeness makes it the definitive tool for assessing closed-loop stability using the Nyquist stability criterion.

Core Concept Explanation

The Nyquist plot is constructed by applying the principle of contour mapping from complex analysis. A closed contour is defined in the s-plane that encircles the entire right-half plane (RHP). This contour, called the Nyquist contour, consists of three segments: the positive imaginary axis from 0 to +j-infinity, a large semicircle of radius R approaching infinity that sweeps from +j-infinity through the right side to -j-infinity, and the negative imaginary axis from -j-infinity back to 0.

If G(s)H(s) has poles on the imaginary axis (for example a pole at s=0 for a type-1 system), the Nyquist contour includes a small semicircular indentation of radius epsilon approaching zero that detours around each such pole to the right. This ensures the contour does not pass through any singularities of G(s)H(s).

Every point on the Nyquist contour in the s-plane is mapped through G(s)H(s) to a corresponding point in the G(s)H(s) complex plane. As the contour is traversed, the image traces out the complete Nyquist plot. The positive frequency portion (jw, w: 0 to infinity) is the standard polar plot. The negative frequency portion is the mirror image of the polar plot about the real axis.

Mathematical Expression

The Nyquist stability criterion states that the number of closed-loop poles in the RHP (Z) equals the number of open-loop poles in the RHP (P) plus the net number of clockwise encirclements (N) of the -1 + j0 point by the Nyquist plot.

Z = N + P

For closed-loop stability, Z must equal zero, which means N = -P. If P = 0 (open-loop stable system), then N must be 0 for stability, meaning the Nyquist plot must not encircle (-1, j0) at all. Counterclockwise encirclements are counted as negative, clockwise as positive.

Practical Understanding

When a system has poles on the imaginary axis, the indentation around those poles must be carefully mapped. For a type-1 system with a pole at s=0, the small semicircular indentation in the s-plane has radius epsilon going from -j-epsilon to +j-epsilon via the right side. This semicircle maps to a large semicircle of radius 1/epsilon in the G(s)H(s) plane, sweeping in the opposite angular direction by the pole order times 180 degrees.

The large semicircle at R approaching infinity generally maps to the origin of the G(s)H(s) plane for proper transfer functions (degree of denominator greater than numerator), so it contributes no encirclements. This simplifies construction significantly for most practical systems.

Given:
G(s) = K / [s(s+2)], H(s) = 1, K = 4
Construct key points and check stability using Nyquist criterion

Why this formula applies:
Type-1 system: pole at origin requires indentation, P = 0 (no RHP open-loop poles)

Formula:
Z = N + P, for stability Z = 0 so N must = 0

Substitution:
G(jw) = 4 / [jw(jw+2)] = 4 / [j2w + jw^2*... ]
G(jw) = 4 / [jw(jw+2)] = 4 / [(-w^2 + j2w)]
Real part: G_re = 4*(-w^2) / (w^4 + 4w^2) = -4/(w^2+4)
Imag part: G_im = -4*2w / (w^2*(w^2+4)) = -8/[w(w^2+4)]

At w -> 0+: G_re -> -1, G_im -> -infinity (locus comes from -j infinity)
At w -> inf: G_re -> 0, G_im -> 0 (locus ends at origin)
Phase crossover: G_im = 0 when w -> infinity only, so locus crosses negative real axis only at w -> 0 (contribution from indentation)

Calculation:
The Nyquist plot does not encircle (-1, j0) for K=4. N = 0.
Z = 0 + 0 = 0: no closed-loop RHP poles.

Final Answer:
System is stable for K = 4. Gain margin = 2/K * something; verify from Routh as GM = infinity for this type.

Exam Tip: For GATE, when the Nyquist plot crosses the negative real axis, check the crossing point value. If the plot crosses at a real value of -1/K_critical, the gain margin equals K_critical times the nominal gain ratio. This directly connects Nyquist analysis to gain margin calculation.

Construction Steps

• Step 1: Identify the system type and check for imaginary axis poles. Design indentation detours for each such pole.
• Step 2: Compute G(jw) for w from 0+ to infinity and plot the positive frequency locus (this is the standard polar plot).
• Step 3: Mirror the positive frequency locus about the real axis to get the negative frequency locus (w from -infinity to 0-).
• Step 4: Map the indentation around each imaginary axis pole. A simple pole at origin maps to a large semicircle of radius infinity sweeping by 180 degrees.
• Step 5: Close the contour if needed for the large semicircle segment (usually maps to origin for proper systems).
• Step 6: Count net clockwise encirclements N of (-1, j0) and apply Z = N + P to determine closed-loop stability.

Quick Revision

• Nyquist contour encloses entire RHP in s-plane; its G(s)H(s) image is the Nyquist plot.
• Stability criterion: Z = N + P where Z = closed-loop RHP poles, N = net CW encirclements of (-1, j0), P = open-loop RHP poles.
• For an open-loop stable system (P=0), closed-loop stability requires N = 0 (no encirclements of -1+j0).
• Negative frequency locus is always the complex conjugate mirror of the positive frequency locus.
• A pole at origin causes the Nyquist contour to indent right, mapping to a large circle sweeping by 180 degrees in GH plane.
• GATE trap: Counterclockwise encirclements are negative N. For open-loop unstable systems (P not zero), counterclockwise encirclements stabilize the system, not destabilize it.