--- title: "Nyquist plot of (s+2)/[s(s+1)(s+3)]" subject: "Control Systems" chapter: 6 key: "semester_3/control_systems:nyquist-s2-s-s1-s3" source: https://www.bctengineer.com/q/semester_3/control_systems/nyquist-s2-s-s1-s3 site: BCT Engineering --- # Nyquist plot of (s+2)/[s(s+1)(s+3)] Control Systems · Chapter 6: Frequency Response Techniques - Asked in: 2070 Chaitra (old course), 10 marks; 2065 Shrawan (old course), 8 marks (asked 2 times) - Video: [#215 Problem on Nyquist plot in control systems || EC Academy](https://www.youtube.com/watch?v=5PEVdDxZWjA) — EC Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/nyquist-s2-s-s1-s3 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-6 ## Question Draw the Nyquist plot for the following open loop transfer function G(s)H(s) = (s+2)/[s(s+1)(s+3)]. ## Short answer No open-loop RHP poles (P=0). Nyquist plot does not cross negative real axis, so -1 is not encircled (N=0). Closed-loop stable with infinite gain margin. ## Answer $$ G(s)H(s) = \frac{s+2}{s(s+1)(s+3)} = \frac{\tfrac23\left(1 + \tfrac{s}{2}\right)}{s(1+s)\left(1 + \tfrac{s}{3}\right)} $$ Open-loop poles: $0, -1, -3$, so **P = 0**. There is one pole at the origin, so the contour is indented there. #### 1. Magnitude and phase $$ |G(j\omega)| = \frac{\sqrt{\omega^2 + 4}}{\omega\sqrt{\omega^2 + 1}\sqrt{\omega^2 + 9}}, \qquad \angle G = -90^\circ + \tan^{-1}\frac{\omega}{2} - \tan^{-1}\omega - \tan^{-1}\frac{\omega}{3} $$ #### 2. Real and imaginary parts $$ G(j\omega) = \frac{(2 + j\omega)\left[-4\omega^2 - j(3\omega - \omega^3)\right]}{16\omega^4 + (3\omega - \omega^3)^2} $$ $$ \text{Re} = \frac{-5\omega^2 - \omega^4}{16\omega^4 + (3\omega - \omega^3)^2}, \qquad \text{Im} = \frac{-6\omega - 2\omega^3}{16\omega^4 + (3\omega - \omega^3)^2} $$ - Im < 0 for every ω > 0, so the plot **never crosses the real axis** at a finite frequency. The phase approaches −180° only as ω → ∞. - Low-frequency asymptote: as ω → 0, Re → $-5/9 = -0.556$. (Equivalently $K\sum T = \tfrac23(1 + \tfrac13 - \tfrac12) = 0.556$.) #### 3. Table (ω > 0) | ω | \|G\| | ∠G | Re | Im | |---|---|---|---|---| | 0⁺ | ∞ | −90° | −0.556 | −∞ | | 0.1 | 6.64 | −94.8° | −0.55 | −6.62 | | 0.5 | 1.21 | −112.0° | −0.45 | −1.12 | | 1 | 0.50 | −126.9° | −0.30 | −0.40 | | 2 | 0.175 | −142.1° | −0.14 | −0.11 | | 5 | 0.036 | −159.5° | −0.03 | −0.01 | | ∞ | 0 | −180° | 0 | 0 | #### 4. Mapping of the full contour - **ω = 0⁺ → +∞:** from $\infty\angle-90^\circ$ (along Re = −0.556) down to the origin, arriving at −180° (third quadrant). - **ω = −∞ → 0⁻:** mirror image in the second quadrant. - **Infinite semicircle:** maps to the origin. - **Small indentation at s = 0:** $G \approx \tfrac23/s$ maps to an infinite arc, clockwise through the right half-plane (from +90° to −90°). ```text .. | . | .. | .. | .. | ... | ... | ..... | -------@---------------------------*****---------- ***** | *** | *** | ** | ** | ** | * | ** | -0.556 0 * w > 0 . w < 0 (mirror) @ point (-1, j0) asymptote Re = -0.556; no negative real crossing ``` #### 5. Stability **Nyquist criterion:** $Z = P + N$, where $P$ = number of open-loop poles in the RHP, $N$ = net **clockwise** encirclements of $(-1, j0)$ by the Nyquist plot of $G(s)H(s)$, and $Z$ = number of closed-loop poles in the RHP. The closed loop is stable only if $Z = 0$. The plot never cuts the negative real axis, so $(-1, j0)$ is not encircled: N = 0, and $Z = 0 + 0 = 0$. The **closed-loop system is stable**. Also, GM = ∞ and PM = 64.8° (at $\omega_{gc}$ = 0.588 rad/s). Check: $s^3 + 4s^2 + 4s + 2 = 0$ gives Routh first column $1, 4, 3.5, 2$, all positive. ✓ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For G(s)H(s) = (s+2)/[s(s+1)(s+3)], what is P and how is the origin treated? A. P = 1; no indentation is needed B. P = 0; the contour is indented around the pole at the origin C. P = 3; all poles are counted D. P = 0; the pole at the origin is ignored Answer: B. Poles are 0, -1, -3, none in the RHP so P = 0, but the pole at the origin lies on the contour and needs a small indentation. (Key point: Open-loop poles and indentation) 2. Why does the plot not cross the negative real axis at a finite frequency? A. Im is negative for all w > 0, and the phase only approaches -180 degrees as w tends to infinity B. The magnitude is below 1 C. The plot is a circle D. Re is positive for all w Answer: A. Im = (-6w - 2w^3)/(...) is always negative for w > 0, so the plot stays in the third quadrant until it reaches the origin. (Key point: Real axis crossing) 3. What is the low-frequency behaviour of the plot as w tends to 0? A. It starts at a finite value 0.667 on the real axis B. It starts at infinity at -90 degrees, asymptotic to the vertical line Re = -0.556 C. It starts at infinity at -180 degrees D. It starts at the origin Answer: B. The pole at the origin gives infinite magnitude at -90 degrees, with Re approaching -5/9 = -0.556. (Key point: Low-frequency asymptote) 4. Given the full plot, what is the stability conclusion? A. Unstable, N = 1 B. Unstable, since P = 1 C. Marginally stable D. Stable: -1 is not encircled, so N = 0 and Z = 0 Answer: D. The plot never reaches the negative real axis before the origin, so (-1, j0) is not encircled; Z = P + N = 0 and the Routh check on s^3 + 4s^2 + 4s + 2 confirms it. (Key point: Stability result) ## Related in Control Systems - [Transfer function from asymptotic Bode plot](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-from-bode-asymptote) — Control Systems, chapter 6, asked 2× - [Nyquist plot of (s+2)/[(s+1)(s−1)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-s2-over-s2-1) — Control Systems, chapter 6, asked 3× - [Nyquist plot of (s+10)/[(s−3)(s+3)] and gain margin](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-s10-s-3-s3) — Control Systems, chapter 6, asked 2× - [Nyquist plot of K/[s(s²+2s+2)] and maximum K](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-k-s-s2-2s-2) — Control Systems, chapter 6 - [Nyquist plot of 2.2/[s(s+1)(s²+2s+2)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Anyquist-22-s-s1-s2-2s) — Control Systems, chapter 6 - [Determining transfer function from Bode plot](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-to-tf) — Control Systems, chapter 6, asked 2× ## Related in other subjects - [DT transfer function; drawing a Bode plot](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adt-transfer-function-bode) — Signals and Systems, chapter 6 - [Open loop versus closed loop control](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aopen-vs-closed-loop) — Embedded System, chapter 7, asked 6× - [Murray loop test for open and short circuit faults](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Amurray-loop-open-short-circuit) — Power System Analysis I, chapter 2, asked 5× - [Open-loop automobile cruise controller design](https://www.bctengineer.com/question.md?key=semester_6%2Fembedded_system%3Aopen-loop-cruise-controller) — Embedded System, chapter 7, asked 3×