--- title: "Bode plot of 1000/[s(1+0.1s)(1+0.001s)]" subject: "Control Systems" chapter: 6 key: "semester_3/control_systems:bode-1000-s-01s-0001s" source: https://www.bctengineer.com/q/semester_3/control_systems/bode-1000-s-01s-0001s site: BCT Engineering --- # Bode plot of 1000/[s(1+0.1s)(1+0.001s)] Control Systems · Chapter 6: Frequency Response Techniques - Asked in: 2081 Bhadra, 7+1 marks - Video: [Bode plot in control system](https://www.youtube.com/watch?v=qnsznGf70eU) — Smart Engineer - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/bode-1000-s-01s-0001s - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-6 ## Question Draw the Bode plot of the unity feedback system with an open loop transfer function G(s) = 1000/[s(1+0.1s)(1+0.001s)]. Also comment on stability. ## Short answer Asymptotes: –20 dB/dec to 10 rad/s, –40 to 1000 rad/s, then –60. ωgc≈ωpc≈100 rad/s, GM≈0.09 dB, PM≈0.06°; system is barely (marginally) stable. ## Answer #### Factors and corner frequencies $$ G(s)=\frac{1000}{s(1+0.1s)(1+0.001s)} $$ - $K=1000$ (60 dB), one pole at origin: initial slope $-20$ dB/dec. - Corner frequencies: $\omega_1=1/0.1=10$ rad/s, $\omega_2=1/0.001=1000$ rad/s. #### Asymptotic magnitude plot | Range | Slope | |---|---| | $\omega<10$ | $-20$ dB/dec | | $10<\omega<1000$ | $-40$ dB/dec | | $\omega>1000$ | $-60$ dB/dec | $$ \begin{aligned} \omega=1:&\ 20\log1000 = 60\ \text{dB}\\ \omega=10:&\ 60-20 = 40\ \text{dB}\\ \omega=100:&\ 40-40 = 0\ \text{dB}\\ \omega=1000:&\ 0-40 = -40\ \text{dB}\\ \omega=10^4:&\ -40-60 = -100\ \text{dB} \end{aligned} $$ #### Phase plot $$ \phi(\omega)=-90^\circ-\tan^{-1}0.1\omega-\tan^{-1}0.001\omega $$ | $\omega$ | 1 | 10 | 50 | 100 | 1000 | $10^4$ | |---|---|---|---|---|---|---| | $\phi$ | $-95.8^\circ$ | $-135.6^\circ$ | $-171.6^\circ$ | $-180^\circ$ | $-224.4^\circ$ | $-264.2^\circ$ | ```text dB 60 |\ -20 40 | \___ w=10 0 |-------\---------------- w | w=100\ -40 -40 | \__ w=1000 | \ -60 deg | -90 |--__ -180 |-------x---------------- | 100 \___ -270 | ---- 1 10 100 1000 1e4 ``` #### Crossover frequencies and margins - **Phase crossover:** $\tan^{-1}0.1\omega+\tan^{-1}0.001\omega=90^\circ \Rightarrow 0.1\times0.001\,\omega^2=1 \Rightarrow \omega_{pc}=100$ rad/s. $$ |G(j100)|=\frac{1000}{100\sqrt{1+100}\sqrt{1+0.01}}=0.990,\quad GM=-20\log0.990=+0.086\ \text{dB} $$ - **Gain crossover:** the asymptote gives 0 dB at $\omega=100$ rad/s; exactly $\omega_{gc}=99.5$ rad/s, where $\phi=-179.94^\circ$. $$ PM=180^\circ-179.94^\circ\approx0.06^\circ $$ #### Comment on stability (1 mark) $\omega_{gc}$ and $\omega_{pc}$ almost coincide, and GM ≈ 0 dB, PM ≈ 0°. The closed-loop system is **just stable (on the verge of instability, practically marginally stable)**: its dominant poles are at $-0.049\pm j99.5$, giving a very lightly damped oscillation at about 100 rad/s. Any small increase in gain makes it unstable. Routh check: $0.0001s^3+0.101s^2+s+1000$: $0.101\times1=0.101>0.0001\times1000=0.1$, so stable by a very small margin. A compensator (e.g. lead) is needed for acceptable relative stability. **Answer: $\omega_{gc}\approx\omega_{pc}\approx100$ rad/s, GM ≈ 0.09 dB, PM ≈ 0.06°: the system is barely (marginally) stable.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For G(s) = 1000/[s(1+0.1s)(1+0.001s)], what is the asymptotic magnitude at w = 1 rad/s? A. 40 dB B. 100 dB C. 30 dB D. 60 dB Answer: D. 20 log 1000 = 60 dB at w = 1 on the -20 dB/dec line. (Key point: Low-frequency gain) 2. What is the slope between w = 10 and w = 1000 rad/s? A. -20 dB/dec B. -40 dB/dec C. -60 dB/dec D. -80 dB/dec Answer: B. After the pole at 10 rad/s the slope is -40 dB/dec until the pole at 1000 rad/s. (Key point: Corner frequencies and slopes) 3. Which equation gives the phase crossover frequency? A. 0.1 x 0.001 x w^2 = 1, so w = 100 rad/s B. 0.1w = 1, so w = 10 rad/s C. 0.001w = 1, so w = 1000 rad/s D. w = 0.1 + 0.001 Answer: A. tan^-1 0.1w + tan^-1 0.001w = 90 degrees requires (0.1)(0.001)w^2 = 1, so w = 100 rad/s. (Key point: Phase crossover frequency) 4. With GM about 0.09 dB and PM about 0.06 degrees, how should the system be described? A. Comfortably stable B. Clearly unstable C. Barely stable, almost marginally stable D. Overdamped Answer: C. Both margins are nearly zero and the crossover frequencies almost coincide, so any slight gain increase causes instability. (Key point: Stability comment) ## Close variants set in other exams - [Bode plot of 40(s+2)/[(2+s+25s²)(1+2s)s]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-40-s2-2-s-25s2) — Control Systems, chapter 6, asked 3× - [Bode plot of 512(s+3)/[s(s²+16s+256)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-512-s3-s2-16s-256) — Control Systems, chapter 6, asked 4× - [Bode plot of 60/[s(s+2)(s+6)] with gain margin](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-60-s-s2-s6) — Control Systems, chapter 6, asked 4× ## Related in Control Systems - [Lead compensator for k/[s(1+0.1s)(1+0.001s)], Kv = 1000](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Alead-k-s-01s-0001s-kv1000) — Control Systems, chapter 7, asked 3× - [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× - [Transfer function with G3 feedforward, H1 and unity loop](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abdr-g1g2g3-h1-unity-2070o) — Control Systems, chapter 3 - [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× - [Transfer function from Bode plot with example](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-to-tf-example) — Control Systems, chapter 6, asked 2× - [GM and PM from Bode plot of 1/[s(0.1s+1)(0.2s+1)]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Abode-gm-pm-1-s-01s) — Control Systems, chapter 6 ## 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