Nyquist Plot - Control Systems - Electrical Engineering (EE) PDF Download

Introduction

• The Nyquist plot is a frequency-domain graphical representation of a complex function obtained by mapping a contour in the s-plane into the complex plane of the function. It is commonly used to assess closed-loop stability from the open-loop frequency response.
• Compared with a polar plot, which shows the locus of G(jω) (or magnitude and phase) for ω from 0 to +∞ only, the Nyquist plot shows the locus for ω from -∞ to +∞. This accounts for both positive and negative frequency contributions and gives the complete mapping required for the Nyquist stability test.
• Let a complex rational transfer function be written as: F(s) = (s - z1)(s - z2) ... (s - zm) / (s - p1)(s - p2) ... (s - pn). The roots z1...zm are the zeroes and p1...pn are the poles of the function. In physical linear systems typically n ≥ m.
• The complex variable is s = σ + jω, so F(s) is complex and may be written F(s) = u(σ,ω) + j v(σ,ω). Every point s where F(s) is analytic maps to a point in the F(s) plane. A closed contour in the s-plane therefore maps to a (generally different) closed contour in the F(s) plane.
• The Nyquist method detects the presence of closed-loop poles in the right-half of the s-plane by relating the open-loop frequency response L(jω) = G(jω)H(jω) to the zeros and poles of 1 + G(s)H(s) that lie inside the chosen s-plane contour.

Contour in the s-Plane

• A contour in the s-plane is a closed path chosen so that the mapped image of the contour by a complex function can be analysed. For Nyquist stability testing the contour normally encloses the entire right-half s-plane (RHP) and follows the imaginary axis from ω = 0 → +∞, a large semicircle in the RHP from +j∞ → -j∞, and then along the imaginary axis from ω = -∞ → 0. The contour avoids singularities on the imaginary axis by small detours if necessary.
• Encircled: A point is said to be encircled by a closed path when the point lies inside the closed path-i.e. the closed path winds around that point. In standard Nyquist discussion, the encirclement is considered with sign: anticlockwise encirclements are taken positive and clockwise negative (this sign convention will be used when applying the argument principle).
• Enclosed: A point is enclosed by the closed path if it lies to the right of the path when the path is traversed in its chosen direction. For Nyquist contours the notion of "to the right" is useful when describing regions contained by clockwise or anticlockwise traversals.

Nyquist Stability Criterion - Principle and Statement

• The closed-loop transfer function with unity feedback is T(s) = C(s)/R(s) = G(s) / (1 + G(s)H(s)) . The characteristic equation is 1 + G(s)H(s) = 0. Zeros of 1 + G(s)H(s) are the closed-loop poles.
• Define the open-loop function L(s) = G(s)H(s). Consider the complex function F(s) = 1 + L(s).
• Argument principle (used in Nyquist derivation): If F(s) is analytic on and inside a closed contour Γ in the s-plane except for P poles inside Γ, then the number of anticlockwise encirclements N of the origin by the mapped contour F(Γ) equals Z - P, where Z is the number of zeros of F(s) inside Γ and P is the number of poles of F(s) inside Γ: N = Z - P.
• Apply this to F(s) = 1 + L(s). If Γ is the Nyquist contour encircling the right-half s-plane, let P be the number of open-loop poles of L(s) inside Γ (i.e. in the RHP) and Z be the number of zeros of F(s) inside Γ (i.e. closed-loop poles in the RHP). The Nyquist plot of L(jω) about the point -1 + j0 gives encirclement count N (anticlockwise positive). From the argument principle: Z = P + N. This is the central Nyquist relation.
• Consequences for closed-loop stability:
  • If Z = 0 (no closed-loop poles in RHP) then the closed-loop system is stable. Hence for stability one must have N = -P (that is, the Nyquist plot must produce N anticlockwise encirclements where N equals -P; equivalently there must be P clockwise encirclements of -1 if P > 0).
  • If the open-loop has no RHP poles (P = 0) then closed-loop stability requires N = 0, i.e. the Nyquist plot must not encircle -1 in the anticlockwise sense.
• The Nyquist test therefore reduces closed-loop stability determination to a counting of encirclements of the point -1 + j0 by the Nyquist plot of the open-loop L(jω), together with knowledge of the number of open-loop RHP poles.

Practical Steps to Construct a Nyquist Plot and Perform Stability Analysis

The construction is normally split into mapping portions of the Nyquist contour separately and then joining the mapped pieces. For a typical rational L(s) this is done in four sections C1-C4; the final Nyquist plot is the union of the mapped sections.

Step 1 - Determine poles and zeros of L(s)

• Identify poles and zeros of L(s) = G(s)H(s). Note poles that lie in the RHP (these count as P). Note poles on the imaginary axis - these are singular points which require small detours in the contour and special care.

Step 2 - Choose the Nyquist contour

• Choose a contour that encloses the entire RHP. Typical contour: along the imaginary axis from j0 to j∞ (C1), a large semicircle in the RHP from j∞ to -j∞ (C2), along the imaginary axis from -j∞ to j0 (C3), plus small semicircles around any imaginary-axis poles (C4 or infinitesimal detours). If no imaginary-axis poles exist the small detours are not required.

Step 3 - Map the contour: sections C1-C4

• Section C1: s = jω with ω from 0 → +∞. The mapped locus is L(jω) for ω ≥ 0 (this is the polar plot for positive frequencies). Practical methods to sketch it include evaluating magnitude and phase versus ω, plotting key frequencies (gain-crossover and phase-crossover), or using component factor contributions (poles/zeros) to sketch magnitude/phase behaviour.
• Section C2: large semicircle in the RHP. For rational L(s) where degree(numerator) ≤ degree(denominator), the contribution of the large semicircle often tends to zero or follows a predictable behaviour. On this arc s = R e^jθ with θ from +90° down to -90°. For high R the dominant s-powers determine the mapping and phase change contributed by the arc equals (n - m)×(θ variation) in degrees where n,m are pole/zero counts - this is helpful in predicting how the Nyquist plot wraps around the origin at large radii.
• Section C3: s = jω with ω from -∞ → 0. For real-coefficient systems L(-jω) = L(jω)* (complex conjugate), so the negative-frequency branch is the mirror image of the positive-frequency branch about the real axis. Thus C3 is the mirror of C1 across the real axis.
• Section C4: small semicircles around any poles on the imaginary axis (if present). Their mapping must be separately evaluated; for a pole at the origin (s = 0) the small detour maps to a circular arc in the L-plane. If no imaginary-axis poles exist the small detours and C4 can be omitted.

Notes on frequency ranges and special cases

• When L(s) contains terms of the form (1 + sT), for a large semicircle s ≈ R e^jθ the factor (1 + sT) ≈ sT if R ≫ 1/T. This approximation simplifies the high-frequency mapping.
• If L(s) contains a pole at the origin (type > 0), the Nyquist contour must avoid the origin with a small semicircle and the mapped image of that small detour must be included in the final Nyquist curve.

Worked Example

Example: Draw the Nyquist plot for the system whose open-loop transfer function is

G(s)H(s) = K / [ s (s + 2) (s + 10) ]

Also determine the range of K for which the closed-loop system is stable.

Solution - Stepwise mapping and stability check

Step 1: Determine poles and zeros of the open-loop.

The transfer function has no finite zeros (numerator constant K), and three poles at s = 0, s = -2, s = -10.

Therefore the system is type 1 (one pole at origin) and order 3. There are no open-loop poles in the RHP, hence P = 0.

Step 2: Map section C1 (s = jω, ω from 0 → +∞).

Write L(jω) explicitly:

L(jω) = K / [ jω (jω + 2) (jω + 10) ]

Separate real and imaginary parts to find crossings and key frequencies. Multiply out denominator to obtain a real and imaginary expression for the complex denominator and hence for L(jω). Evaluating the frequency at which the imaginary part of L(jω) becomes zero gives the phase-crossover frequency (where the locus crosses the real axis).

Set imaginary part = 0. Simplifying gives the cubic terms; the crossing condition reduces to:

ω (1 - 0.05 ω²) = 0

Hence positive solution is

ω = sqrt(20) ≈ 4.472 rad/s (the phase-crossover frequency).

Evaluate the real part of L(jω) at ω = 4.472 to get the real axis crossing point:

L(jω) (real crossing) = -0.00417 K

Thus the Nyquist locus for ω ≥ 0 starts at -j∞ (phase -90° for small ω, because of the pole at origin) and as ω increases it approaches the real axis and crosses near -0.00417 K on the real axis.

Step 3: Map section C2 (large semicircle in RHP).

For high |s| the factors (1 + sT) may be approximated by sT where appropriate; substituting dominant terms shows the mapping of the large semicircle contributes a predictable rotation. For this third-order system with a pole at the origin the contribution from the large semicircle is consistent with the system order and does not create additional encirclements around -1 in typical finite K ranges. A standard sketch shows the mapped arc passing at large radius and returning to the mirror branch.

Step 4: Map section C3 (ω from -∞ → 0).

This branch is the mirror image of C1 about the real axis because L(-jω) = [L(jω)]*. Therefore the negative frequency part completes the Nyquist plot by mirroring the positive frequency locus.

Step 5: Map small semicircle at origin (C4) due to pole at s = 0.

Because there is a pole at the origin, we must detour around s = 0 with a small semicircle. Its mapped image corresponds to a circular arc in the L-plane. For this example, that arc does not introduce encirclements of -1 for the practical K range to be determined, but it must be represented in the complete Nyquist sketch.

Step 6: Determine K for which Nyquist plot passes through (-1 + j0).

The real crossing for positive ω occurred at Re{L(jω)} = -0.00417 K. Set this equal to -1 to find the limiting K where the curve passes through -1.

Equate and solve:

-0.00417 K = -1

K = 1 / 0.00417 ≈ 240

Step 7: Combine the mapped sections to form the complete Nyquist plot and interpret encirclements.

Stability conclusion using Z = P + N and P = 0:

If K < 240 then the nyquist plot does not encircle -1 (n="0)." therefore z="0" + 0="0" and the closed-loop has no rhp poles ⇒ closed-loop 

If K > 240 then the Nyquist plot crosses and produces encirclements of -1 (the example shows two clockwise encirclements for K sufficiently greater than 240 in the given sketch). Clockwise encirclements count negative in the anticlockwise convention, so N < 0 and z="P" + n becomes positive, indicating closed-loop rhp poles ⇒ closed-loop 

Therefore the closed-loop system is stable for

0 < K < 240.

Advantages of the Nyquist Plot

• Determines closed-loop stability directly from the open-loop frequency response without solving the characteristic polynomial.
• Capable of handling systems with time delay (a factor that appears as e^{-sτ} and produces an unbounded phase lag), where root-locus techniques are less convenient.
• Provides a visual way to use and interpret Bode and polar plot information (gain and phase margins can be read from Nyquist behaviour).
• Gives the frequency response of the open-loop and indicates how many closed-loop poles lie in the RHP once open-loop RHP poles are counted.
• Useful for assessing relative stability (how close the system is to instability) via encirclements and margins.

Limitations and Cautions

• Requires careful handling of poles on the imaginary axis and of infinite semicircles; small detours must be included and properly mapped.
• Interpreting encirclements requires correct accounting of the sign (anticlockwise positive); incorrect sign use leads to wrong stability conclusions.
• Nyquist is not primarily a tool to find exact pole locations - it counts zeros of 1 + L(s) in the RHP but does not give their exact values. For exact pole positions, root-locus or direct root computation is required.
• Construction can be algebraically intensive for high-order systems, so practical use often relies on numerical plotting tools or Bode/Nyquist plotting in software (MATLAB, Python control packages) for accuracy.