Fault tree analysis (FTA) — top-down, AND - OR gates: Visual walkthrough

3.6.30 · D2 · Physics › Spacecraft Structures & Systems Engineering › Fault tree analysis (FTA) — top-down, AND - OR gates

Kuch bhi shuru karne se pehle, ek word jo hum har line pe use karenge:

Step 1 — Ek basic event ko area ke roop mein draw karo

KYA. Poore tree ki sabse simple cheez lo: ek component, ya toh working ya failed. Iske failure probability ko $P_{1}$ kaho.

KYUN. Tree mein upar ki har cheez aise events ko combine karke bani hai. Agar hum ek single event ko picture nahi kar sakte, toh combination ko bhi nahi kar sakte. Isliye hum leaf se shuru karte hain.

PICTURE. Figure dekho. Poore square ka area $1$ hai (possibilities ka poora box). Hum $P_{1}$ width ka ek amber strip shade karte hain — woh shaded area exactly component 1 ke fail hone ka chance hai. White part, width $1 - P_{1}$ , woh chance hai ki yeh survive karta hai.

Step 2 — Ek square par do independent events

KYA. Ek doosra component lao jiske failure probability $P_{2}$ hai, pehle se independent (ek ka outcome doosre ke baare mein kuch nahi batata). Step 1 ka naming rule is naye part ke liye copy karo: $q_{2} = 1 - P_{2}$ isko iska survival number maano.

KYUN. Ek gate mein hamesha kam se kam do inputs hote hain. Unhe combine karne ke liye pehle unhe ek saath lay out karna hoga taaki hum saare overlap regions dekh sakein.

PICTURE. Hum strip ko poore grid mein badal dete hain. Component 1 ki failure vertical amber band hai (width $P_{1}$ ); component 2 ki failure horizontal cyan band hai (height $P_{2}$ ). Kyunki woh independent hain, square saaf tarike se chaar rectangles mein split ho jaata hai, aur har rectangle ka area uske side lengths ka product hai. Yeh sides-ka-multiplication hi poori wajah hai ki independence humein probabilities multiply karne deti hai.

Step 3 — AND gate = woh corner jahan sabhi fail hote hain

KYA. AND gate "failure" tab output karta hai jab saare inputs fail ho jaate hain. Hamare grid par yeh exactly ek rectangle hai: top-right corner.

KYUN. Spacecraft mein yeh redundancy hai: solar aur battery dono ko marna padega tabhi power jaayegi. Hum dekhna chahte hain ki woh corner kitna chota hai, kyunki chota = safe.

PICTURE. Sirf overlap rectangle shaded hai. Iska width $P_{1}$ hai, height $P_{2}$ hai, toh iska area — AND-gate probability — product hai. $1$ se neeche ke do numbers ka multiplication hamesha kisi bhi ek se chota result deta hai: yeh shrinking corner hi wajah hai ki redundancy kaam karta hai.

Step 4 — OR gate = sab kuch siwaaye all-safe corner ke

KYA. OR gate "failure" output karta hai agar koi bhi input fail ho. Yeh poora square minus woh ek white corner hai jahan sab survive kar gaye.

KYUN. Yeh single point of failure hai: solar power fail ho jaati hai agar deployment jam ho ya cells degrade ho ya tracking fail ho — koi bhi ek alone fatal hai. Koi redundancy nahi.

PICTURE. Poore grid ko shade karo, phir bas bottom-left "both work" rectangle kaat lo. Jo bacha woh OR probability hai. Surviving corner ka area $q_{1} q_{2} = (1 - P_{1}) (1 - P_{2})$ hai, toh shaded remainder $1$ minus that hai.

Step 5 — Solar branch compute karo (teen ka OR)

KYA. Step 4 ko teen real solar failures par apply karo: deploy jam $0.001$ , degradation $0.01$ , tracking $0.005$ . Ab gate mein $n = 3$ inputs hain, toh yeh pehli baar hai jab hum OR formula do se zyada ke saath use kar rahe hain.

KYUN. Yeh teeno ek OR gate ko feed karte hain, toh koi bhi ek solar power ko dooba deta hai. Aur upar jaane se pehle humein yeh number chahiye.

PICTURE — teen factors picture par kaise rehte hain. Do inputs ke saath "all survive" region ek rectangle tha (area = do side-lengths multiply). Teen inputs ke saath yeh ek 3D box ban jaata hai (volume = teen side-lengths multiply): survival ke liye part 1 aur part 2 aur part 3 ka kaam karna zaroori hai, toh hum teeno survival widths $0.999 \times 0.99 \times 0.995$ multiply karte hain. Ek mushkil cube draw karne ki jagah, figure teen survival factors ko stacked slabs ke roop mein rakhta hai — white heights ko upar se neeche padhna aur multiply karna exactly yahi hai "3D box ki har axis par baari-baari chalo". Failure area paane ke liye us product ko $1$ se subtract karo.

Step 6 — Tree par chadhna: gates feeding gates

KYA. Solar OR gate ka output aur battery AND gate ka output top-level AND gate ke do inputs ban jaate hain.

KYUN. Ek fault tree recursive hota hai — ek gate ka answer uske upar wale gate ke liye bas ek basic-event-sized number hai. Yahi top-down structure hai jo parent ne promise kiya tha, bottom-up evaluate kiya jaata hai.

PICTURE. Tree ko dubara draw kiya gaya hai jisme har gate ki computed probability uske paas likhi hai, upar ki taraf flow karti hai: leaves → branch gates → top event.

Step 7 — Edge & degenerate cases (koi gap mat chhodna)

KYA. Har input ko uski extremes par push karo aur check karo ki pictures abhi bhi sahi behave karti hain.

KYUN. Jo formula tum sirf "nice" numbers ke liye trust karte ho woh ek aisa formula hai jo $P = 0$ ya $P = 1$ par surprise dega. Hum corners test karte hain.

PICTURE. Chaar mini-panels: ek input $0$ par, ek input $1$ par, aur dono gate types.

Ek AND input $0$ hai (ek component jo kabhi fail nahi hota): failure corner ki width zero hai → $P_{AND} = 0$ . Perfect redundancy branch ko immune bana deta hai. ✔
Ek OR input $1$ hai (ek component jo hamesha fail hota hai): "all survive" corner mein zero-width factor hai → $P_{OR} = 1 - 0 = 1$ . Ek guaranteed single-point-failure poore branch ko guarantee se fail karata hai. ✔
Ek AND input $1$ hai: $P_{AND} = 1 \times P_{2} = P_{2}$ — certain part drop out ho jaata hai, sirf uncertain wala bachta hai. ✔
Ek OR input $0$ hai: $(1 - 0) = 1$ ek factor of $1$ contribute karta hai, toh ek never-failing part OR mein kuch nahi badalta. ✔

Ek-picture summary

Upar ki sab cheez ek diagram mein collapse ho jaati hai: AND = tiny overlap corner (multiply, risk shrinks), OR = whole box minus survival corner (risk grows), aur gates stack hote hain taaki har answer uske upar wale gate ko feed kare. Unit square poore tree ke peeche ka single mental model hai.

Recall Feynman retelling — simple shabdon mein bolo

Ek square picture karo jo "jo bhi ho sakta hai" ko represent karta hai, total area ek. Har tarike ke liye slivers shade karo jisme ek part toot sakta hai. Agar gate AND hai — matlab dono parts ko toota chahiye tumhe hurt karne ke liye — tum sirf us chhote overlapping corner ki parwah karte ho jahan slivers cross karte hain; do widths multiply karo aur corner chota hai, jo exactly wajah hai ki backup rakhna tumhe safe rakhta hai. Agar gate OR hai — matlab koi bhi part alone hurt karta hai — tum poora square shade karte ho siwaaye us ek corner ke jahan kuch nahi toota; woh leftover bada hai, jo exactly wajah hai ki single weak link dangerous hai. Real tree karne ke liye, yeh do corner-games bottom mein karo, har answer ko ek number ke roop mein likho, aur use upar bhejo jaise woh ek nayi part ho, jab tak top na pahunch jao. Solar teen weak links tha ek OR mein (lagbhag 1.6%), battery do backups tha ek AND mein (0.04%), aur top ne kaha tumhe dono systems ko marne chahiye — ek AND — toh multiply karo aur tumhe ek million mein lagbhag six chances milte hain. Yahi poora method hai: do pictures, poore upar tak repeat karte hue.

Recall

Unit-square picture par, AND-gate probability kaun sa corner hai? ::: Woh overlap corner jahan har input ek saath fail hota hai — area $P_{1} P_{2} \dots$ . OR gate $1 - \prod (1 - P_{i})$ kyun use karta hai $P_{i}$ add karne ki jagah? ::: Add karne se overlapping failure corners double-count ho jaate hain; single "all survive" corner subtract karna saara overlap bookkeeping avoid kar deta hai. AND input exactly $0$ branch ke saath kya karta hai? ::: Poore AND output ko zero kar deta hai — ek never-failing redundant part branch ko immune bana deta hai. Ek OR input exactly $1$ kya karta hai? ::: OR output ko $1$ force karta hai — ek guaranteed single-point failure branch ko guarantee se fail karata hai. Agar do inputs ek component share karte hain toh kaun sa extra step chahiye? ::: Shared component ko apna basic event banao; independence ab hold nahi hoti — dekho Common Cause Failure Analysis.

See also: Boolean Algebra and Logic Design · Minimum Cut Sets · Reliability Block Diagrams · Single Point Failure Review · Failure Modes and Effects Analysis (FMEA) · Spacecraft Redundancy Architectures · Monte Carlo Reliability Simulation