Foundations — Block diagram algebra

3.5.28 · D1 · Physics › Guidance, Navigation & Control (GNC) › Block diagram algebra

Is page par kuch bhi assume nahi kiya gaya. Pehle "series-combine blocks" ya "apply the feedback formula" karne se pehle, tumhe exactly pata hona chahiye ki ek block kya hai, letter $s$ ka matlab kya hai, arrow kya carry karta hai, aur ek box kyun multiply karta hai. Hum har symbol ko ground up se build karte hain, ek ek karke, pehle waala samajhne ke baad hi agle par jaate hain.

0. Woh picture jise hum decode kar rahe hain

Parent topic mein sab kuch sirf char tarah ke marks se bana hai: ek arrow, ek box, ek circle, aur ek dot. Chalo inhe ek ek karke meet karte hain.

Figure dekho. Flow ko left se right trace karo: ek signal enter karta hai left arrow se, box ke through jaata hai, aur right par ek naye signal ki tarah nikalta hai. Yahi ek block ki poori kahani hai. Ab hum har piece ko naam dete hain.

1. Arrow — ek signal

Picture: ek directional wire. Pipe mein paani, cable mein current, line par voltage — arrow ko physically kya hai uski chinta nahi; yeh sirf ek number jo change ho sakta hai carry karta hai.

Kyun topic ko iske zaroorat hai: parent ke har rule mein ("output $=G\cdot$input", "$E=R-B$") signals ke baare mein statement hai. Arrows ke bina multiply, add, ya feed back karne ke liye kuch nahi hoga.

Hum kaafi named signals se milenge. Parent in letters ka use karta hai:

Har ek sirf ek kaam wala arrow hai. Inhe yaad rakho; jaise hi hum loop draw karte hain, yeh wapas aate hain.

2. Variable $s$ — "Laplace ki duniya"

Pehle hum yeh batayein ki box kya karta hai, hume uske andar ke letter ko decode karna hoga, kyunki box mein "$G(s)$" hai aur woh $s$ bahut kaam kar raha hai.

Yeh tool kyun aur koi nahi? Time ki duniya mein, kisi signal ko physical system ke through bhejna matlab convolution karna hai, ek integral jo input ko time mein smear karta hai — genuinely painful arithmetic. Laplace transform is sawaal ka jawaab deta hai "kya koi aisa viewpoint hai jahan woh smearing ordinary multiplication ban jaaye?" aur jawaab haan hai: woh viewpoint variable $s$ hai. Hum $s$ precisely isliye choose karte hain kyunki yeh ugly operation ko $\times$ mein convert kar deta hai.

Is topic ko use karne ke liye tumhe Laplace transform compute karne ki zaroorat nahi. Tumhe sirf ek fact maanna hai jo yeh deta hai: through-a-system $=$ multiply-by-$G(s)$.

3. Box — ek transfer function $G(s)$

Ab box samajh aata hai.

Picture: ek gain-machine. Jo bhi value left arrow par flow karke aati hai use $G(s)$ se multiply karke right arrow par bheja jaata hai. Section 0 ke figure mein, agar input $X$ hai toh output $G(s)X$ hai.

$G(s)$ kyun, sirf number $G$ kyun nahi? Ek plain number ka matlab hoga "hamesha 3 se multiply karo, chahe signal kitni bhi fast wiggle kare." Real systems slow vs. fast changes par alag respond karte hain (ek heavy motor fast commands mein lag karta hai). Multiplier ko $s$ par depend karake ek box un saari speed-dependent behaviours ko ek saath encode kar sakta hai. $G(s)$ ke andar kya hota hai yeh Transfer functions ka kaam hai.

Figure box ko multiplier ki tarah dikhata hai: input $X$ (cyan) enter karta hai, amber label $G(s)$ use scale karta hai, aur output $G(s)X$ (white) nikalta hai. Yeh single multiplication poore topic ka atom hai.

4. Circle — ek summing junction

Picture: wires ka ek meeting-point jo running total output karta hai. Parent ki key equation $E=R-B$ aise circle se seedhi padhi jaati hai: reference $R$ $+$ ke saath aata hai, fed-back signal $B$ $-$ ke saath aata hai, aur circle unka difference $E$ ugalta hai.

Kyun topic ko iske zaroorat hai: yahan subtraction control ka dil hai. Hum chahte hain ki error $E$ = "hum kitne galat hain?" Ek summing junction jisme feedback par $-$ hai woh galti measure karta hai taaki system correct kar sake.

Figure mein signs dhyan se dekho. Wahi circle add kar sakta hai ($E=R+B$) ya subtract ($E=R-B$) — sirf arrow ke paas chhota symbol badalta hai. Parent mein "positive vs. negative feedback" ka har distinction usi ek sign mein hai.

5. Dot — ek take-off (pickoff) point

Picture: ek splitter. Ek pipe mein paani do identical pipes mein fan out hota hai; dono wahi same value carry karte hain jo dot par pahunchi thi. Agar dot par signal $A$ hai, toh dot se nikalne wali har wire bhi $A$ carry karti hai.

Kyun topic ko iske zaroorat hai: feedback aur parallel paths dono "same signal do taraf jaata hai" se shuru hote hain. Take-off woh jagah hai jahan signal reuse hota hai. Aur — yeh ek favourite trap hai — copy woh value padhta hai bilkul usi exact location par. Dot ko alag jagah move karo aur copy alag value padh lega. Yahi single fact parent ke Rule 4 mein "take-off ko block se aage move karne par $\frac{1}{G}$ lagta hai" ki wajah hai.

6. Marks ko saath jodna — ek full loop

Ab parent ke bade formula ka har symbol define ho gaya hai. Chalo inhe us single-loop diagram mein assemble karte hain jise parent "the big one" kehta hai aur picture se equation padhte hain.

• Circle $E=R-B$ deta hai (Section 4).
• Forward box $G$ $Y=GE$ deta hai (Section 3).
• Feedback box $H$ $B=HY$ deta hai — same rule, alag box.
• $Y$ se circle tak wire ek take-off dot (Section 5) par shuru hoti hai: yeh $Y$ copy karta hai taaki output bahar bhi jaye aur fed back bhi ho sake.

Exactly parent ki tarah substitute karne par closed-loop transfer function $\frac{Y}{R}=\frac{G}{1+GH}$ milta hai. Dhyan do ki hum koi naya symbol use nahi kiya — arrow, box, circle, dot, aur $s$ se milne wala multiplier fact kaafi tha. Yahi is foundations page ka point hai.

7. Prerequisite map

Equipment checklist

Connections

• Parent topic — full algebra & rules
• Laplace transform — "through-a-block $=$ multiply" fact deta hai.
• Transfer functions — har box $G(s)$ ke andar actually kya hota hai.
• Feedback control loops — jahan assembled loop use hota hai.
• Signal flow graphs & Mason's gain formula — in marks ko redraw karne ka ek alternative.
• Op-amp gain — same closed-loop $G/(1+GH)$ ki kahani.
• Stability & characteristic equation — denominator $1+GH$ jo humne abhi banaya woh poles set karta hai.