Visual walkthrough — Block diagram algebra

3.5.28 · D2 · Physics › Guidance, Navigation & Control (GNC) › Block diagram algebra

Hum sirf yeh ek cheez assume karte hain: ek arrow ek signal hai jo travel kar raha hai, aur ek block ek box hai jisse tum multiply karte ho. Baaki sab hum khud banate hain.

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Step 1 — Ek arrow, ek box: "multiply" kaisa dikhta hai

KYA: Ek single signal $X$ draw karo jo ek box mein enter kare jis par $G$ likha hai, aur ek signal $Y$ jo usse bahar nikle.

KYUN: Koi bhi loop aane se pehle, humein bilkul pakka hona chahiye ki ek block kya karta hai. Ek block koi delay nahi hai, koi filter nahi hai jise tum wiggles ki tarah imagine karo — hamare algebra ke liye yeh ek instruction hai: incoming arrow par jo number hai use box mein likhe number se multiply karo.

PICTURE:

$Y=G\cdot X$

• $X$ — input arrow par signal (wire ke us point par ek value).
• $G$ — box ke andar transfer function; multiplier.
• $Y$ — output arrow par signal, jo $G$ times $X$ ke barabar hai.

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Step 2 — Summing junction: ek circle jo add aur subtract karta hai

KYA: Woh chhota circle introduce karo jo do arrows andar leta hai aur ek arrow bahar deta hai. Har incoming arrow par $+$ ya $-$ sign hota hai.

KYUN: Feedback ka matlab hai "output ko command se compare karo." Compare karna = subtract karna. Humein ek picture-object chahiye jo subtract kare, aur woh hai summing junction.

PICTURE:

$E=R-B$

• $R$ — reference (jo hum system ko karne ka command dete hain); $+$ ke saath andar aata hai.
• $B$ — fed-back signal (jo actually hua uski ek copy); $-$ ke saath andar aata hai.
• $E$ — error, hum kitne door hain: command minus reality.

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Step 3 — Take-off point: ek signal ko free mein copy karna

KYA: Output wire par ek dot lagao. Us dot se ek doosra arrow branch out karta hai. Copying se value nahi badlti — dot ke baad dono arrows exactly $Y$ hi carry karte hain.

KYUN: Kuch feed back karne ke liye, pehle humein use tap karna padta hai bina disturb kiye. Take-off (pickoff) point wahi tap hai.

PICTURE:

• Dot $Y$ carry karne wali wire par baitha hai.
• Seedha jaane wala arrow abhi bhi $Y$ carry karta hai (real output).
• Branch arrow bhi $Y$ carry karta hai — yahi copy hai jo hum feedback path mein bhejenge.

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Step 4 — Feedback path: copy ko $H$ se hokar wapas bhejo

KYA: $Y$ ki branched copy ko ek doosre box $H$ se route karo, phir summer ke $-$ terminal mein signal $B$ ki tarah daalo.

KYUN: Hum raw output ko rarely wapas feed karte hain — hum uski ek measurement feed back karte hain (ek sensor, ek scaling). Woh measurement box hai $H$, feedback transfer function.

PICTURE:

$B=H\cdot Y$

• $Y$ — feedback box mein enter karta copied output.
• $H$ — feedback block (jaise kisi sensor ka gain).
• $B$ — jo bahar nikalta hai, summer ke $-$ terminal ki taraf jaata hai.

Ab loop physically complete hai: $R$ aur $B$ summer par milte hain, error $E$ $G$ ko drive karta hai, $G$ $Y$ produce karta hai, $Y$ copy hota hai aur $H$ se hokar $B$ ban ke wapas aata hai.

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Step 5 — Ek signal ko poore loop mein trace karo

KYA: Teen facts likhao jo picture pehle se force karti hai, us order mein jis order mein signal travel karta hai.

KYUN: Poora diagram ab teen equations hai. Agar hum internal signals $E$ aur $B$ ko eliminate kar sakein, toh sirf $R$ aur $Y$ bachenge — aur unka ratio wahi hai jo hum chahte hain.

PICTURE: wahi loop, ab har wire labelled ke saath:

$$\underset{\text{summer (Step 2)}}{\underbrace{E=R-B}}\underset{\text{block (Step 1)}}{\underbrace{Y=GE}}\underset{\text{feedback (Step 4)}}{\underbrace{B=HY}}$$

• Pehli equation: summer ka rule.
• Doosri: forward block ka rule.
• Teesri: feedback block ka rule.

Hamare paas teen equations hain aur teen unknown internal signals hain, plus do external wale. Solve karne ke liye bilkul sahi hai.

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Step 6 — Substitute karke internal signals ko khatam karo

KYA: $B=HY$ ko $E=R-B$ mein daalo, phir woh $E$ ko $Y=GE$ mein daalo.

KYUN: $E$ aur $B$ private wires hain jo box ke andar hain — bahar ka user unhe kabhi nahi dekhta. Unhe eliminate karne se sirf woh relation bachta hai jo humara command ($R$) aur jo hum paate hain ($Y$) ke beech hai.

$$Y=GE=G(R-B)=G(R-HY)$$

term by term:

• $G(R-HY)$ — block error ko multiply karta hai.
• $R$ — command.
• $HY$ — fed-back measurement jo subtract ho rahi hai.

$G$ distribute karo:

$$Y=GR-GHY$$

• $GR$ — open-loop response: $Y$ kya hota agar loop cut hota.
• $GHY$ — woh correction jo loop har baar subtract karta rehta hai, kyunki $Y$ ek baar per lap apne aap ko influence karta hai.

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Step 7 — $Y$ collect karo: loop-gain appear hota hai

KYA: Har $Y$ ko left side mein le jaao aur factor karo.

KYUN: $Y$ dono sides par appear hota hai kyunki output apne aap ko feed karta hai. $Y$ terms ko gather karna ek self-referential loop ko ek clean solvable line mein badal deta hai.

$$Y+GHY=GR⟹Y(1+GH)=GR$$

• $1$ — signal jo bina kisi feedback trip ke seedha ek baar jaata hai (woh "$Y$ akela").
• $GH$ — loop gain: ek signal ko ek poore lap ke baad kitna multiply kiya jaata hai ($G$ forward $\times$ $H$ back).
• $1+GH$ — passthrough plus ek round trip.

Divide karo:

$$\boxed{\frac{Y}{R}=\frac{G}{1+GH}}(\text{negative feedback})$$

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Step 8 — Har case: signs, unity, aur degenerate limits

KYA: Humein koi bhi scenario undraw nahi chhodna chahiye. Har special case walk karo.

KYUN: Jo formula tum stress-test nahi kar sakte, woh samjha nahi. Yahan har sign aur har extreme hai.

Positive feedback. Agar summer $B$ ko add kare ($E=R+B$), toh Step 6 dobara karo: $Y=G(R+HY)=GR+GHY$, toh $Y(1-GH)=GR$: $\frac{Y}{R}=\frac{G}{1-GH}(\text{positive feedback}).$ Sirf denominator mein sign badla — kyunki round-trip ab reinforce karta hai oppose karne ki jagah.

Unity feedback ($H=1$): sensor ek plain wire hai, $B=Y$. Tab $\frac{Y}{R}=\frac{G}{1+G}.$

Koi loop nahi ($H=0$): feedback path cut hai, $B=0$, toh $E=R$ aur $Y=GR$. Formula agree karta hai: $\frac{G}{1+0}=G$. ✔ Sanity check: loop open hone par tum Step 1 ka plain block hi paate ho.

Bahut bada forward gain ($G\to \infty$): $\frac{Y}{R}=\frac{G}{1+GH}\stackrel{G\to \infty }{\to }\frac{1}{H}.$ Output $G$ ki parwah karna band kar deta hai aur sirf $H$ par depend karta hai. Yahi reason hai ek op-amp's ka enormous, sloppy open-loop gain ek precise closed-loop gain ban jaata hai jo feedback resistors se set hota hai.

Denominator zero ho jaaye ($1+GH=0$): ratio blow up hota hai — ek finite command infinite output drive karta hai. Woh equation, $1+GH=0$, characteristic equation hai; iske roots loop ke poles hain. Yeh woh boundary hai jiske ek taraf controller settle karta hai aur doosri taraf runaway ho jaata hai.

PICTURE — ek formula ke chaar roop:

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Ek-picture summary

Ek diagram, ek substitution chain, ek formula. Ise top to bottom padho: signal $R$ andar aata hai, apna error paata hai, $G$ se multiply hota hai, copy hota hai, $H$ se measure hota hai, $B$ banke wapas aata hai — aur "$Y$ ek baar per lap apne aap ko feed karta hai" ka algebra exactly woh $1+GH$ hai jo neeche hai.

Recall Feynman: poora walkthrough simple words mein

Ek thermostat socho. Tum ek temperature command karte ho ($R$). Room ki real temperature measure hoti hai aur wapas bheji jaati hai ($B$). Summer argue karta hai: "command minus reality" hi error hai ($E$). Heater block $G$ hai — woh error ko actual heat mein badalta hai ($Y=GE$). Ab woh sneaky part: jo heat tumne abhi banayi woh nayi reality ban jaati hai, phir measure hoti hai, aur wapas aake error ko shrink karti hai. Toh $Y$ secretly apne aap par depend karta hai. Jab tum woh dependence honestly likhte ho — $Y$ barabar hai uski open-loop value $GR$ minus feedback ka ek lap $GHY$ — aur $Y$'s ko gather karte ho, toh milta hai $Y(1+GH)=GR$. "$1$" tum ho jo seedha ek baar se guzre; "$GH$" ek poora trip around hai. Divide karo, aur loop ek single number $\frac{G}{1+GH}$ mein tame ho jaata hai. $G$ ko bahut bada kar do aur machine apni khud ki strength bhool jaati hai aur sirf sensor $1/H$ ki maanti hai — feedback ka yahi poora trick hai.

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Connections

• Parent: Block diagram algebra — woh rules jo yeh page derive karta hai.
• Transfer functions — har box mein kya rehta hai.
• Laplace transform — kyun ek box ka matlab "multiply" hai.
• Feedback control loops — jahan $\frac{G}{1+GH}$ use hota hai.
• Signal flow graphs & Mason's gain formula — wahi result bina diagram surgery ke.
• Op-amp gain — hardware mein $G\to \infty \Rightarrow 1/H$ limit.
• Stability & characteristic equation — denominator $1+GH=0$.

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Quick self-check