Block diagram algebra

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

A block diagram ek equation ki picture hoti hai. Block-diagram algebra bas woh rules hain jinse hum us picture ko rewrite kar sakte hain bina uski equation ko badle. Ise master karo aur tum kisi bhi tangled GNC loop ko ek transfer function mein collapse kar sakte ho.

Rule 1 — Blocks in series (cascade)

KYA HAI: Do blocks ek ke baad ek.

KAISE / derive karo: Maano input $= X$ . Pehle block ke baad signal hai $G_{1} X$ . Use doosre mein daalo: output $= G_{2} (G_{1} X) = G_{1} G_{2} X$ .

Yeh step kyun? Multiplication associative hoti hai, toh scalar transfer functions ke liye order matter nahi karta: $G_{1} G_{2} = G_{2} G_{1}$ .

Rule 2 — Blocks in parallel

KYA HAI: Same input $X$ do blocks mein split (take-off point) hoti hai jinke outputs ek summer par milte hain.

KAISE / derive karo: Top branch deta hai $G_{1} X$ , bottom deta hai $G_{2} X$ . Summer add karta hai: $Y = G_{1} X \pm G_{2} X = (G_{1} \pm G_{2}) X$ .

Rule 3 — Feedback loop (sabse bada wala)

KYA HAI: Forward path $G (s)$ , feedback path $H (s)$ , negative summer.

KAISE / first principles se derive karo:

Error (summer output): $E = R - B$ .
Output: $Y = G E$ .
Fed-back signal: $B = H Y$ .

$B$ substitute karo: $E = R - H Y$ . Phir $Y = GE = G (R - H Y) = GR - G H Y$ . $Y$ collect karo: $Y + G H Y = GR \Rightarrow Y (1 + G H) = GR$ .

Rule 4 — Blocks ko move karna (equivalence moves)

Yeh $Y / R$ ko unchanged rakhte hain. Har ek ko signals equal rakhne ki condition se derive karo.

Worked Example 1 — Cascade forward path wale loop ko collapse karo

System: $R \to [summer, -] \to G_{1} \to G_{2} \to Y$ , feedback $H$ .

Step 1: Forward path ko series-combine karo: $G = G_{1} G_{2}$ . Kyun? Rule 1 do cascaded blocks ko ek mein turn karta hai.
Step 2: Feedback formula apply karo: $\frac{Y}{R} = \frac{G _{1} G _{2}}{1 + G _{1} G _{2} H}$ . Kyun? Ab yeh standard single-loop form hai.

Maano $G_{1} = \frac{2}{s}$ , $G_{2} = \frac{1}{s + 3}$ , $H = 1$ : $\frac{Y}{R} = \frac{2/ [ s ( s + 3 )]}{1 + 2/ [ s ( s + 3 )]} = \frac{2}{s ^{2} + 3 s + 2} = \frac{2}{( s + 1 ) ( s + 2 )} .$ Yeh step kyun? Fractions clear karne ke liye top aur bottom ko $s (s + 3)$ se multiply karo — pure algebra hai, koi nayi physics nahi.

Worked Example 2 — Take-off point move karo

Forward: $R \to [-] \to G \to Y$ . Ek take-off $G$ ke baad $H$ se feedback karta hai. Hum chahte hain ki feedback $G$ se pehle tap ho.

Step 1: Abhi $B = H Y = H G E$ .
Step 2: Agar take-off ko $G$ se pehle move karein, ab woh $E$ (= $G$ ki input) padhta hai, $Y = GE$ nahi. $B$ same rakhne ke liye moved branch ko $G$ se multiply karna hoga: naya feedback block $= G H$ . Kyun? Hum ne $G$ insert kiya taaki jo multiplication kho gayi thi woh recover ho sake.
Check: $B = G \cdot H \cdot E$ — pehle jaisa hi. ✔

Worked Example 3 — Do loops (inner + outer)

Inner loop: forward $G_{2}$ , feedback $H_{2}$ . Outer loop: forward $G_{1}$ phir inner block, feedback $H_{1}$ .

Step 1 (inner reduce karo): $G_{inn er} = \frac{G _{2}}{1 + G _{2} H _{2}}$ . Kyun? Hamesha pehle innermost loop collapse karo — outer rules ke liye wahan ek single block chahiye.
Step 2 (series): forward path $= G_{1} G_{inn er} = \frac{G _{1} G _{2}}{1 + G _{2} H _{2}}$ .
Step 3 (outer loop): $\frac{Y}{R} = \frac{\frac{G _{1} G _{2}}{1 + G _{2} H _{2}}}{1 + \frac{G _{1} G _{2}}{1 + G _{2} H _{2}} H _{1}} = \frac{G _{1} G _{2}}{1 + G _{2} H _{2} + G _{1} G _{2} H _{1}} .$ Yeh step kyun? Nested fraction clear karne ke liye top aur bottom ko $(1 + G_{2} H_{2})$ se multiply karo.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho ek water slide hai. Har slide piece tumhari speed ek khaas amount badhaata hai (yeh ek block hai — woh tumhari speed ko multiply karta hai). Agar do slides connect hain, tum dono speeds multiply karte ho. Ek splitter tumhe do lanes mein copy karta hai; jahan lanes milti hain wahan tum flows add karte ho. Ek feedback loop woh hota hai jab neeche ka kuch paani waapas upar pump hota hai aur incoming paani se subtract hota hai — yeh tank ko overflow hone se rokta hai. Magic rule yeh hai: poori cheez ka behaviour = (forward slide) ÷ (1 + poore loop ka trip). "+1" bas tum ho, seedha neeche, ek baar.

Flashcards

Series blocks

G_{1}, G_{2}

combine hokar banate hain

G_{1} G_{2}

(multiply)

Parallel blocks combine hokar banate hain

G_{1} \pm G_{2}

(add, sign summer se)

Forward

G

, feedback

H

(negative) ke liye Closed-loop TF

\frac{G}{1 + G H}

Unity-feedback closed-loop TF

\frac{G}{1 + G}

"Loop gain" kya hai?

G H

— woh factor jo ek signal loop ke ek poore chakkar mein gain karta hai

Positive feedback mein denominator ban jaata hai

1 - G H

Ek summing junction ko block

G

ke past downstream move karo: kya correction?

moved branch ko

G

se multiply karo

Ek take-off point ko block

G

ke past downstream move karo: kya correction?

us branch ko

1/ G

se multiply karo

Nested loops reduce karne ka order

inside-out (innermost loop pehle)

s

-domain mein blocks multiply kyun karte hain?

time mein convolution = Laplace domain mein multiplication

1 + G H

mein

1

ka physical meaning

signal ka direct passthrough (koi feedback trip nahi)

Connections

Transfer functions — woh objects jo har block ke andar rehte hain.
Laplace transform — kyun blocks multiply karte hain.
Feedback control loops — Rule 3 ka application.
Signal flow graphs & Mason's gain formula — in moves ka ek algebra-free alternative.
Op-amp gain — same $G / (1 + G H)$ closed-loop math.
Stability & characteristic equation — denominator $1 + G H = 0$ poles set karta hai.

Core objects