Foundations — Fault tree analysis (FTA) — top-down, AND - OR gates

3.6.30 · D1 · Physics › Spacecraft Structures & Systems Engineering › Fault tree analysis (FTA) — top-down, AND - OR gates

Fault tree banana ya padhna shuru karne se pehle (dekho parent topic), tumhe ek toolbox chahiye. Parent note mein freely words use hote hain jaise probability, product, independent events, aur symbols jaise $P$, $\prod$, $\times$. Yahan hum inhe absolute zero se earn karte hain, ek aisi order mein jahan har item sirf pehle wale par depend karta hai.

1. "Event" kya hota hai?

Tasveer: socho ek light switch. Woh ya toh ON hai (event hua) ya OFF hai (nahi hua). Koi "half-on" nahi hota.

Yeh topic ko kyun chahiye: ek fault tree poori tarah se inhi on/off boxes se bani hoti hai. Parent note mein har "Solar Fail", "Cell1", "HGA Fail" ek aisa hi switch hai. Failures ko combine karne se pehle, hume agree karna hoga ki har failure ek clean yes/no cheez hai.

2. Probability — 0 aur 1 ke beech ek number

Symbol $P$ bas "ki probability" words ka shorthand hai. Toh $P(\text{Battery Fail})=0.02$ padhte hain "battery fail hone ki probability 0.02 hai".

Yeh topic ko kyun chahiye: parent note mein har basic event ($P=0.001$, $P=0.05$, …) is ruler par ek point hai. Poora "quantitative analysis" inhi numbers par arithmetic hai.

3. Complement — "woh hone ki chance jo NAHI hoga"

Tasveer: poore ruler ki length $1$ hai. Agar "hota hai" wala part $P$ length ka slice leta hai, toh bacha hua slice — "nahi hota" — zaroor $1-P$ hoga.

Yeh topic ko kyun chahiye: parent note mein OR-gate formula poori tarah complements se bana hai — "1 minus kuch bhi fail na hone ki chance". Jab tak $1-P$ obvious nahi lagta, tum woh formula nahi padh sakte.

4. Independent events — jab ek ka doosre par koi asar nahi

Tasveer: do alag dice. Pehle die par 6 aana doosre die ki odds nahi badalta. Woh alag alag worlds mein rehte hain.

Yeh topic ko kyun chahiye: parent note mein AND aur OR dono probability formulas mein ek hidden footnote hai — "independent events ke liye". Agar events independent nahi hain, toh woh formulas galat hain. Isliye yeh concept har calculation ki raksha karta hai.

5. Probabilities ka multiplication — independent events ka "AND"

Yeh AND gates ke liye sabse important idea hai.

Symbol $\times$ ka matlab ordinary multiplication hai. Tasveer: ek rectangle. Agar width $P(A)$ hai aur height $P(B)$ hai, toh area hai dono ke hone ki chance.

Yeh topic ko kyun chahiye: yahi AND-gate formula hai. Parent note mein har AND gate apne inputs ko multiply karta hai.

6. Product symbol $\prod$ — "poori list ko multiply karo"

Ise piece by piece padhte hain: ${\prod }_{i=1}^n{P}_i=P_1\times P_2\times \cdots \times P_n$

• $i$ ek counter hai (page number ki tarah). Yeh bottom value se shuru hota hai aur 1-1 karke badhta hai.
• Symbol ke neeche $i=1$: counting 1 se shuru karo.
• Upar $n$: rukk jao jab counter $n$ tak pahunche (items ki total sankhya).
• $P_i$: "$i$-vi probability" — $i=1,2,3,\dots$ baari baari daalo.

Yeh topic ko kyun chahiye: parent note mein AND-gate formula ($\prod P_i$) aur OR-gate formula ($\prod (1-P_i)$) dono mein yeh symbol use hota hai. Iska matlab bas "list ko multiply karo" hai — kuch aur nahi.

7. OR gate formula — sab kuch ek saath

Ab sections 3, 4, 5, 6 ke har piece milkar parent note ka sabse scary-looking formula banate hain.

Ise inside-out padho, un tools se jo ab tumhare paas hain:

1. $1-P_i$ — chance ki input $i$ survive kare (complement, §3).
2. $\prod (1-P_i)$ — chance ki sabhi inputs ek saath survive karein (independents ko multiply karo, §5–6).
3. $1-(\text{woh})$ — chance ki woh sab survive nahi karte, yaani kam se kam ek fail hota hai (complement phir se, §3).

Yeh topic ko kyun chahiye: parent note isi se $P_{\text{Solar}}=0.016$ compute karta hai. Ab tum usmein har symbol samajhte ho.

8. Tree ki tasveer khud — gates, branches, top aur bottom

Tasveer: ek family tree ulta bana hua. Disaster upar baitha hai; uske causes neeche branch karte hain; sabse chhote causes neeche leaves hain. Neeche padhna = "yeh kyun ho sakta hai?". Upar compute karna = "toh disaster kitna likely hai?".

Yeh topic ko kyun chahiye: yahi woh object hai jo poora method produce karta hai. AND/OR ke peeche ki algebra ke liye dekho Boolean logic, aur usi logic ko alag tarah draw karne ke liye dekho Reliability Block Diagrams.

Prerequisite map

Equipment checklist

Khud test karo — daayaa side chhupao aur upar se jawab do.