Bode plot — magnitude and phase vs frequency

3.5.41 · Physics › Guidance, Navigation & Control (GNC)

1. Sinusoids kyun aur $s=j\omega$ kyun?

YEH KYUN hota hai (derive karo). Ek system lo jiska transfer function $G(s)$ hai aur input $u(t)=\cos \omega t=\text{Re}\{e^{j\omega t}\}$ hai. Complex exponential $e^{j\omega t}$ kisi bhi LTI system ka eigenfunction hota hai:

$y(t)=G(j\omega )e^{j\omega t}$

Complex number $G(j\omega )$ ko polar form mein likho:

$G(j\omega )=|G(j\omega )|e^{j\varphi },\varphi =\angle G(j\omega )$

Tab

$y(t)=|G(j\omega )|e^{j(\omega t+\varphi )}\Rightarrow \text{Re}\{y\}=|G(j\omega )|\cos (\omega t+\varphi )$

Toh Bode plot literally complex-valued function $G(j\omega )$ ki ek picture hai — magnitude aur angle do alag graphs mein split ho jaate hain.

2. Decibels kyun aur log frequency kyun?

Derivation. Maano $G=G_1{G}_2{G}_3$. Tab

$|G|=|G_1||G_2||G_3|$

$20{\log }_{10}$ lo:

$\underset{\text{dB}}{\underbrace{20{\log }_{10}|G|}}=20{\log }_{10}|G_1|+20{\log }_{10}|G_2|+20{\log }_{10}|G_3|$

Aur phase seedha add hoti hai (product ka angle = angles ka sum):

$\angle G=\angle G_1+\angle G_2+\angle G_3$

Log-frequency axis kyun: control systems kai decades span karte hain (0.01 rad/s slosh → 1000 rad/s structural modes). Log spacing sabko dikhata hai, aur — bonus — simple factors ke asymptotes straight lines ban jaate hain jinka slope dB/decade mein hota hai.

3. Building blocks (har asymptote derive karo)

Koi bhi rational $G(j\omega )$ in factors mein decompose hoti hai. Yeh 4 seekh lo aur tum kuch bhi sketch kar sakte ho (80/20 core).

(a) Constant gain $K$

$|K{|}_{\text{dB}}=20{\log }_{10}K\text{ (flat horizontal line)},\angle K=0^{\circ }$

(b) Integrator $\frac{1}{s}=\frac{1}{j\omega }$

$|G{|}_{\text{dB}}=20{\log }_{10}\frac{1}{\omega }=-20{\log }_{10}\omega$ Yeh ek straight line hai jiska slope $-20$ dB/decade hai ($\omega$ mein har 10× par 20 dB drop). $\angle \frac{1}{j\omega }=\angle \frac{1}{e^{j90^{\circ }}}=-90^{\circ }\text{ (constant)}$

(c) First-order pole $\frac{1}{1+s/{\omega }_c}$, corner (break) frequency ${\omega }_c$

$|G(j\omega )|=\frac{1}{\sqrt{1+(\omega /{\omega }_c{)}^2}}$

Asymptotes kyun:

• Low freq $\omega \ll {\omega }_c$: term $\to 1$, toh $|G{|}_{\text{dB}}\to 0$ (flat).
• High freq $\omega \gg {\omega }_c$: $|G|\approx {\omega }_c/\omega$, toh slope $-20$ dB/decade.
• $\omega ={\omega }_c$ par: $|G|=1/\sqrt{2}$, yaani $20{\log }_{10}(1/\sqrt{2})=-3.01\text{ dB}$ — famous −3 dB corner.

Phase: $\angle G=-\arctan (\omega /{\omega }_c)$: low par $0^{\circ }$, ${\omega }_c$ par $-45^{\circ }$, high par $-90^{\circ }$.

(d) First-order zero $(1+s/{\omega }_c)$

Pole ka mirror image: slope +20 dB/decade corner ke upar, phase $0^{\circ }\to +45^{\circ }\to +90^{\circ }$ tak badhti hai.

4. Worked examples

5. Common mistakes (steel-manned)

6. Active recall

7. Flashcards

Connections

• Transfer function — woh $G(s)$ jo hum $s=j\omega$ par evaluate karte hain.
• Nyquist plot — wohi $G(j\omega )$ complex plane mein plot kiya; stability ke liye −1 ko encircle karta hai.
• Stability margins — gain & phase margins seedhe Bode se padhe jaate hain.
• Lead-lag compensator — crossover ke paas Bode plot ko shape karke design kiya jaata hai.
• Low-pass filter — ek first-order pole; iska −3 dB corner iska cutoff hai.
• Laplace transform — jahan se $s$, poles, aur zeros aate hain.
• Loop shaping — GNC design method jo poori tarah Bode magnitude par driven hai.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps