Block diagram algebra: Question bank — Hinglish deep dive

3.5.28 · D5 · Physics › Guidance, Navigation & Control (GNC) › Block diagram algebra

Shuru karne se pehle, ek vocabulary anchor — taaki neeche koi bhi word undefined na rahe:

$R$ — reference input: woh signal jo hum loop mein daalte hain (the "command", jaise "is altitude par jao"). Yeh woh arrow hai jo sabse pehle summer mein enter karta hai.
$E$ — error signal: summer ka output, $E = R - B$ , yaani command aur jo feed back ho raha hai uske beech ka gap. Socho "woh gap jo controller ko close karna hai."
$B$ — fed-back signal: woh signal jo feedback block summer ko wapas bhejta hai, $B = H Y$ .
$Y$ — output: woh signal jo forward block se nikalti hai, $Y = GE$ . Yeh woh cheez hai jo hum actually measure ya move karte hain.
$G$ , $H$ blocks hain: aisi boxes jo unme enter hone wale signal ko $s$ ki ek function se multiply karti hain. Socho "ek slide jo tumhari speed ko scale karta hai."
Summer ek circle hai jo usme enter hone wale arrows ko add ya subtract karta hai.
Take-off point ek dot hai jo ek signal ko kayi wires mein copy karta hai (copying se kuch nahi badalta).
Loop gain $= G H$ : woh total factor jo ek signal loop mein ek baar jaane aur wapas aane par uthata hai.

Neeche di gayi picture un charon symbols ko real wires se fix karti hai — jab bhi koi derivation $R$ , $E$ , $B$ , ya $Y$ ka naam le, iske paas wapas aao.

Kyun numerator kabhi nahi badalta jab feedback positive ho jaata hai, yeh dekhne ke liye, do summers ko side by side dekho.

Spot the error

"Inner aur outer loops? Main pehle outer loop reduce karunga kyunki yeh main control loop hai." — flaw dhundho

Outer feedback formula ko apna forward path already ek single block hona chahiye, lekin woh path abhi bhi un-reduced inner loop chhupa raha hai. Tumhe inside-out kaam karna hoga: pehle inner loop ko ek block mein collapse karo (figure s05).

"Cascade forward path

G_{1}, G_{2}

with feedback

H

: closed-loop

= \frac{G _{1} G _{2}}{1 + G _{2} H}

." — isko pakdo

Loop gain poora round trip hai

G_{1} G_{2} H

, sirf

G_{2} H

nahi. Correct denominator hai

1 + G_{1} G_{2} H

; dono forward blocks loop ke andar hain.

"Unity feedback ka matlab

H = 1

hai, isliye closed-loop TF bas

\frac{G}{2}

hai." — isko pakdo

Tum

G = 1

substitute nahi kar sakte;

G

ek function of

s

hai, number 1 nahi. Unity feedback

H = 1

set karta hai jisse

\frac{G}{1 + G}

milta hai, jo

\frac{G}{2}

nahi hai jab tak

G

exactly 1 na ho.

"Ek take-off ko

G

ke downstream move karte hue, main branch ko

G

se multiply karta hoon match karne ke liye." — isko pakdo

Ulta hai. Downstream tap already

G A

read karta hai (bahut bada), isliye tum

G

se divide karte ho, yaani

\frac{1}{G}

insert karo.

G

se multiply karna woh rule hai jab ek summer ko

G

ke aage move karo, take-off ke liye nahi.

"Parallel branches dono same wire se start hote hain, isliye mujhe take-off point ki zaroorat nahi." — isko pakdo

Chahiye hoga — ek single wire bina take-off dot ke legally do blocks mein fan out nahi ho sakta jo copy ko mark kare. Yeh woh primitive hai jo ek signal ke do jagah feed hone ki permission deta hai.

"Loop gain

G H

stability ko affect nahi kar sakta; sirf numerator poles matter karte hain." — isko pakdo

Stability denominator

1 + G H = 0

se set hoti hai (characteristic equation), jo puri tarah loop gain se bani hai. Numerator poles set nahi karta.

Why questions

Blocks apne inputs ko multiply kyun karte hain, add kyun nahi karte?

Kyunki ek LTI system time mein convolution se kaam karta hai, aur Laplace transform convolution ko

s

mein multiplication mein convert kar deta hai. Isliye "ek block se guzarna" literally hai "by

G (s)

multiply karna."

Nested loops ko inside out reduce kyun karna padta hai?

Har feedback reduction ke liye zaroori hai ki uska forward path pehle se ek single transfer function ho. Un-reduced inner loop ko wrap karne wala outer loop us condition ko violate karta hai, isliye pehle innermost collapse karo.

Feedback "one-plus-loop-gain se divide" kyun karta hai loop gain subtract karne ki jagah?

Kyunki fed-back signal

Y

ke khud proportional hai:

Y = G (R - H Y) = GR - G H Y

- G H Y

ko cross move karne se

Y + G H Y = GR

milta hai, phir

Y (1 + G H) = GR

factor hota hai, isliye

\frac{Y}{R} = \frac{G}{1 + G H}

(1 + G H)

Y

terms ka ek collect-and-factor hai — kabhi bhi gains ka subtraction nahi.

Wahi

\frac{G}{1 + G H}

formula Op-amp gain ka dil kyun hai?

Ek op-amp ka enormous, poorly-controlled open-loop gain

G

hota hai; feedback

H

wrap karne se yeh ek stable, resistor-set closed-loop gain

\frac{G}{1 + G H} \approx \frac{1}{H}

tak divide ho jaata hai jab

G H ≫ 1

. Same block algebra, alag hardware.

Ek poore feedback loop ko reduce hone ke baad ek equivalent block kyun treat kar sakte hain?

Kyunki block algebra us equation ko kabhi nahi badalta jise diagram encode karta hai — yeh sirf picture ko rewrite karta hai. Reduced single block ka identical input-output relation

Y / R

hota hai.

Hum kabhi kabhi Mason's gain formula ko in moves ke upar kyun prefer karte hain?

Kyunki tangled multi-loop diagrams mein equivalence moves error-prone ho jaate hain; Mason's formula seedha graph ke paths aur loops se answer read karta hai bina kuch slide kiye.

Edge cases

Closed-loop TF kya hoga agar loop gain

G H \to \infty

(bahut bada open-loop gain) ho?

\frac{G}{1 + G H} \to \frac{1}{H}

— output sirf feedback path se govern hota hai,

G

se independent. Yahi exactly reason hai kyun bade

G

wale op-amps clean,

H

-controlled gain dete hain.

\frac{G}{1 - G H}

(positive feedback) ka kya hoga jab

G H \to 1

Denominator

\to 0

, isliye closed-loop gain blow up karta hai — system instability/oscillation ke edge par hai. Yeh positive feedback ki resonance/runaway boundary hai.

Ek parallel pair ka equivalent block kya hoga jab

G_{2} = - G_{1}

ek adding summer mein ho?

G_{1} + G_{2} = G_{1} - G_{1} = 0

— branches cancel ho jaate hain aur equivalent block zero hai; koi bhi signal output tak nahi pahunchta. Ek degenerate lekin perfectly legal result.

Agar feedback block

H = 0

ho to loop kya reduce karta hai?

\frac{G}{1 + G \cdot 0} = \frac{G}{1} = G

— bilkul koi feedback nahi, isliye tum open-loop forward block wapas paate ho, jaise expected hota hai jab return path cut ho.

Series-equivalent block kya hoga agar chain mein ek block

G = 1

(ek pure wire) ho?

Yeh 1 ka factor contribute karta hai:

G_{1} \cdot 1 \cdot G_{2} = G_{1} G_{2}

. Ek unity block ek identity element hai — yeh signal ko unchanged pass through karta hai.

Kya tum ek summing junction ko take-off point ke past bina correction ke move kar sakte ho?

Nahi — tumhe dono par har signal value preserve karni hogi. Aise rearrangements ko compensating blocks ya duplicated summers chahiye; hamesha dummy input se test karte hue insist karo ki har wire apni exact value rakhe.

Connections

Transfer functions — woh $G (s)$ , $H (s)$ objects jo har block ke andar rehte hain; upar har trap asal mein ek statement hai is baare mein ki yeh kaise multiply aur combine karte hain.
Laplace transform — woh reason hai kyun "ek block se guzarna" ka matlab "multiply karna" hai, kyunki yeh time-domain convolution ko $s$ -domain multiplication mein convert karta hai (pehle Why question ka jawaab).
Feedback control loops — Rule 3 ka engineering context; is page par $R$ / $E$ / $B$ / $Y$ derivation exactly woh loop hai jo woh systems close karte hain.
Signal flow graphs & Mason's gain formula — woh move-free alternative jo Spot-the-error section mein take-off/summer sliding traps ko sidestep karta hai.
Op-amp gain — woh hardware jahan $G H \to \infty \Rightarrow 1/ H$ edge case actually stable amplifiers banane ke liye use hota hai.
Stability & characteristic equation — explain karta hai kyun denominator $1 + G H = 0$ (numerator nahi) poles set karta hai, last Spot-the-error item justify karta hai.

Question bank — Block diagram algebra

True or false — justify