Exercises — Gain margin, phase margin — stability margins

KYA karte hain: seedha definition mein plug karo. GM woh factor hai jisse tum loop gain ko multiply kar sakte ho jab tak $|L|$ dangerous phase par $1$ tak na pahunche, isliye yeh wahan ki magnitude ka reciprocal hai. $\text{GM}=\frac{1}{0.25}=4.$ dB mein: decibel sirf magnitude ratio ka $20{\log }_{10}$ hai (ek compressed ruler taaki bade aur chote numbers ek axis par fit ho sakein — wohi ruler jo Bode plots use karta hai). ${\text{GM}}_{\text{dB}}=20{\log }_{10}(4)\approx 12.04\text{dB}.$ Sanity check: GM $>1$ (positive dB) ⟹ stable margin. $-1$ ko touch karne se pehle tum gain ko chaar guna kar sakte ho.

KYA aur KYUN: PM current phase se danger angle $-180^{\circ }$ tak ka gap hai. Yeh angle khud nahi hai (classic trap — neeche dekho). $\text{PM}=180^{\circ }+(-120^{\circ })=60^{\circ }.$ Positive ⟹ stable, aur $60^{\circ }$ healthy band $30^{\circ }$–$60^{\circ }$ ke top par baitha hai.

Jhooth. Gain margin phase crossover par hoti hai ($\angle L=-180^{\circ }$), kyunki sirf wahan loop already $-1$ ki direction mein point kar raha hota hai, isliye use reach karne ke liye sirf ek pure gain boost bacha hota hai. (PM woh hoti hai jo $|L|=1$ par padhi jaati hai.)

KYUN $K$ drop ho jaata hai: ek constant gain $K>0$ magnitude scale karta hai lekin zero phase add karta hai. Phase crossover sirf phase equation se define hota hai, isliye $K$ yahan invisible hai. Phase: har factor ek angle contribute karta hai. Pure integrator $1/s$ $-90^{\circ }$ deta hai; denominator mein har $(s+a)$ $-\arctan (\omega /a)$ deta hai. $\angle L=-90^{\circ }-\arctan \omega -\arctan \frac{\omega }{2}=-180^{\circ }.$ To $\arctan \omega +\arctan \frac{\omega }{2}=90^{\circ }$. Jab do arctangents $90^{\circ }$ mein sum hote hain, to unke tangents reciprocals hote hain, yaani $\omega \cdot \frac{\omega }{2}=1$: $\frac{{\omega }^2}{2}=1\Rightarrow {\omega }_{pc}=\sqrt{2}\approx 1.414\text{rad/s}.$ Figure s01 mein phase curve dekho jo exactly $\sqrt{2}$ par $-180^{\circ }$ cross karti hai.

KYA: ${\omega }_{pc}=\sqrt{2}$ par magnitude. Magnitude factor-by-factor multiply hoti hai: $|L|=\frac{K}{\omega \sqrt{{\omega }^2+1}\sqrt{{\omega }^2+4}}.$ ${\omega }^2=2$ par: $\sqrt{{\omega }^2+1}=\sqrt{3},\sqrt{{\omega }^2+4}=\sqrt{6},\omega =\sqrt{2}$. $|L|=\frac{4}{\sqrt{2}\cdot \sqrt{3}\cdot \sqrt{6}}=\frac{4}{\sqrt{36}}=\frac{4}{6}=\frac{2}{3}.$ $\text{GM}=\frac{1}{2/3}=\frac{3}{2}=1.5,{\text{GM}}_{\text{dB}}=20{\log }_{10}(1.5)\approx 3.52\text{dB}.$ Parent note se comparison (jahan $K=1$ ne GM $=6$, yaani $15.6$ dB diya tha): $K$ ko chaar guna karne se GM exactly factor $4$ se shrink ho gaya ($15.6-12.04\approx 3.5$ dB). Gain aur margin directly trade off karte hain.

Rule of thumb: $\zeta \approx \frac{{\text{PM}}^{\circ }}{100}$ PM $\sim 60^{\circ }$ tak ke liye. $\zeta \approx \frac{51.8}{100}\approx \mathrm{0.52.}$ $0.5$ ke paas damping ratio ka matlab hai moderate, well-controlled overshoot — dekho Damping ratio and overshoot. Standard second-order system ke liye, $\zeta =0.5$ roughly $16\%$ overshoot deta hai, ek fast lekin wild nahi response.

KYA: marginal instability ka matlab hai curve exactly $-1$ ko touch karti hai, to $|L(j{\omega }_{pc})|=1$ with $K=K_{\max }$. Problem 2.2 se, ${\omega }_{pc}=\sqrt{2}$ par, $|L|=K/6$. Isko $1$ set karne par: $\frac{K_{\max }}{6}=1\Rightarrow K_{\max }=6.$ Interpretation: $K=6$ par loop $\omega =\sqrt{2}$ rad/s par steady oscillation sustain karta hai. Koi bhi $K>6$ ⟹ GM $<1$ ⟹ unstable. Yeh exactly woh gain hai jahan Root locus branches imaginary axis cross karte hain, aur Nyquist stability criterion curve $-1$ se guzarti dikhegi.

KYUN delay dangerous hai: ek pure delay $e^{-j\omega \tau }$ ki magnitude $1$ hoti hai (yeh kabhi gain nahi badalta) lekin phase $-\omega \tau$ add karta hai — ek phase lag jo frequency ke saath badhta hai. Yeh directly phase margin mein khaata hai bina magnitude plot par koi warning ke. Yeh Time delay and Padé approximation ki kahani hai. PM ko radians mein convert karo: $40^{\circ }\times \frac{\pi }{180}=0.698$ rad. Max delay: gain crossover par delay ka phase lag margin se zyada nahi hona chahiye: ${\omega }_{gc}{\tau }_{\max }=\text{PM (rad)}\Rightarrow {\tau }_{\max }=\frac{0.698}{2}\approx 0.349\text{s}.$ $0.349$ s se zyada hone par extra lag phase ko unity gain par $-180^{\circ }$ se aage push kar deta hai ⟹ PM negative ho jaata hai ⟹ unstable.

Loop B kaafi zyada robust hai. Gain margin sirf ek direction mein safety measure karta hai (pure gain scaling toward $-1$). Ek loop comfortable GM rakhte hue $-1$ ke paas se kisi alag angle se guzar sakta hai — woh closeness hi PM capture karta hai. Loop A ka PM $=15^{\circ }$ ⟹ $\zeta \approx 0.15$ ⟹ heavy ringing aur large overshoot; Nyquist stability criterion plot $-1$ ko skim karta hai. Loop B ka PM $=50^{\circ }$ ⟹ $\zeta \approx 0.5$ ⟹ smooth response. Moral (parent mistake #2 concrete banaya gaya): tumhe dono margins healthy chahiye; akela koi bhi jhooth bol sakta hai.

Pehle phase condition set karo (yeh ${\omega }_{gc}$ fix karta hai $K$ ki value se independent agar hum ek target angle demand karte hain): Gain crossover par humein chahiye $\angle L(j{\omega }_{gc})=-180^{\circ }+45^{\circ }=-135^{\circ }$. $\angle L=-90^{\circ }-\arctan \frac{\omega }{2}=-135^{\circ }\Rightarrow \arctan \frac{\omega }{2}=45^{\circ }\Rightarrow \frac{\omega }{2}=1.$ To ${\omega }_{gc}=2$ rad/s. Ab us frequency par $|L|=1$ enforce karo $K$ pin down karne ke liye: $|L|=\frac{K}{\omega \sqrt{{\omega }^2+4}}=\frac{K}{2\sqrt{8}}=\frac{K}{2\cdot 2.828}=1.$ $K=2\sqrt{8}=4\sqrt{2}\approx \mathrm{5.657.}$ Check: $\zeta \approx \text{PM}/100=0.45$ — nicely damped, small overshoot. Yeh ek mini PID controller tuning step hai: humne ek pure gain choose kiya phase-margin spec hit karne ke liye.

${\omega }_{pc}$ nikalo: $\angle L=-90^{\circ }-\arctan (\omega /2)=-180^{\circ }$ ke liye $\arctan (\omega /2)=90^{\circ }$ chahiye, yaani $\omega \to \infty$. Interpret karo: phase sirf $-180^{\circ }$ ko approach karta hai jab $\omega \to \infty$; yeh actually finite frequency par kabhi nahi pahunchta. Jab $\omega \to \infty$, $|L|\to 0$. $\text{GM}=\frac{1}{|L(j\infty )|}=\frac{1}{0}=\infty .$ Matlab: infinite gain margin — tum $K$ ko kisi bhi factor se multiply kar sakte ho aur yeh two-pole loop kabhi unstable nahi hoga (iska phase $-180^{\circ }$ tak bilkul nahi pahunch sakta). Yahan sari design freedom phase margin mein hai. Dono Nyquist shapes side by side ke liye Figure s02 dekho.

Gain-margin constraint. Problem 2.2 ke pattern se, $|L(j\sqrt{2})|=K/6$, to $\text{GM}=6/K$. Require $\text{GM}\ge 2$: $6/K\ge 2\Rightarrow K\le 3.$ $K=2$ par PM check karo. Pehle ${\omega }_{gc}$ nikalo: $|L|=1$ solve karo: $\frac{2}{\omega \sqrt{{\omega }^2+1}\sqrt{{\omega }^2+4}}=1.$ Numerically ${\omega }_{gc}\approx 0.749$ rad/s (VERIFY mein verified). Wahan phase: $\angle L=-90^{\circ }-\arctan (0.749)-\arctan (0.749/2).$ $\arctan (0.749)=36.83^{\circ },\arctan (0.3745)=20.53^{\circ }.$ $\angle L=-90-36.83-20.53=-147.36^{\circ },\text{PM}=180-147.36=32.6^{\circ }.$ PM $=32.6^{\circ }<40^{\circ }$ fail karta hai! Bhaale $K=2$ gain-margin test pass karta hai (GM $=3=9.5$ dB), phir bhi phase-margin spec violate karta hai. Conclusion: phase margin pehle bind karta hai. Humein $K$ aur neeche karna hoga. Yeh ek textbook Loop shaping insight hai: gain aur phase specs opposite directions mein pull kar sakte hain, aur tum tighter wale ko honour karte ho.

KYA: hum $K$ ko neeche scan karte hain jab tak PM $40^{\circ }$ tak na badh jaaye. $K$ kam karna magnitude curve ko neeche kheenchta hai, ${\omega }_{gc}$ ko kam frequency par move karta hai jahan phase kam negative hoti hai ⟹ larger PM. Numerically, PM $=40^{\circ }$ near $K\approx 1.54$ par hota hai (VERIFY wahan PM $\approx 40^{\circ }$ confirm karta hai). Final range (dono specs, GM $\le 3$ aur PM $\ge 40^{\circ }$): $0<K\lesssim \mathrm{1.54.}$ Gain-margin ceiling $K\le 3$ kabhi active limit nahi hota — phase margin hume roughly $1.54$ par cap karta hai. Ek robust practical pick jaise $K=1$ (PM $\approx 45^{\circ }$, GM $=6$) aaram se andar baithta hai.

1.1 GM $=4=12.04$ dB · 1.2 PM $=60^{\circ }$ · 1.3 phase crossover. 2.1 ${\omega }_{pc}=\sqrt{2}$ · 2.2 GM $=1.5=3.52$ dB · 2.3 $\zeta \approx 0.52$. 3.1 $K_{\max }=6$ · 3.2 ${\tau }_{\max }\approx 0.349$ s · 3.3 Loop B robust. 4.1 $K=4\sqrt{2}\approx 5.657,{\omega }_{gc}=2$ · 4.2 GM $=\infty$. 5.1 GM $K\le 3$ par bind karta hai lekin PM $K=2$ ($32.6^{\circ }$) par fail hoti hai · 5.2 usable $K\lesssim 1.54$.