Solve complex resistor networks
Scenario
Agar ek raste par `2` toll booth hain, aur aage chal kar rasta do mein bat jata hai.
Mixed circuit ko solve karne ke liye kahan se shuru karna chahiye?
Mental model
Complex circuits ek puzzle ki tarah hain. Andar se bahar ki taraf solve karte hain.
Pehle sabse obvious chote loops ko identify karo: kya yeh do resistors pakka series mein hain ya parallel mein? Unhe ek single equivalent resistor se replace karo. Phir naye simplified circuit ko dubara solve karo.
Explanation
Practical problems mein aksar circuits na toh purely series hote hain, na purely parallel. Dono ka mixture hote hain. Ise solve karne ka koi ek formula nahi hai, balki step-by-step approach chahiye.
Rule `1`: Circuit diagram ko dhyan se dekho aur un resistors ko dhoondo jinke beech kahin koi branch/joint nahi hai. Woh pakka series mein hain. Unka `Rs` nikal kar unhe ek single resistor se replace kar do.
Rule `2`: Un resistors ko dhoondo jinke dono ends ek hi common point par jude hain. Woh parallel mein hain. Unka `Rp` nikal kar unhe replace kar do.
Rule `3`: Is process ko tab tak repeat karte raho jab tak poora circuit ek single resistance `R_eq` na ban jaye. Iske baad aap battery ka voltage (`V`) use karke total current (`I = V/R_eq`) nikal sakte hain.
Worked example
Problem: How can three resistors of resistances `2 ohm`, `3 ohm`, and `6 ohm` be connected to give a total resistance of (a) `4 ohm`, (b) `1 ohm`?
- (a) `4 ohm` laane ke liye: `3` aur `6` ko parallel mein jodo.
- Phir us `2 ohm` ko pehle wale `2 ohm` ke saath series mein laga do.
- (b) `1 ohm` laane ke liye: teeno ko parallel mein jod do.
- Total equivalent nikalte hain.
Answer: (a) `3 ohm` & `6 ohm` in parallel, connected in series with `2 ohm`. (b) All three in parallel.
Problem: In a circuit, `R1 = 10 ohm` and `R2 = 40 ohm` are in parallel. They are connected in series to a combination of `R3 = 30 ohm`, `R4 = 20 ohm`, and `R5 = 60 ohm` which are also in parallel. Find total resistance.
- Pehle `R1` aur `R2` ka parallel combination `Rp1` nikalte hain.
- Phir `R3`, `R4`, `R5` ka parallel combination `Rp2` nikalte hain.
- Dono equivalent blocks series mein hain, inko add karte hain.
Answer: `18 ohm`
Key points
- Identify clear sub-networks: Always look for resistors that are unambiguously in series (no branches between them) or in parallel (connected to the exact same two nodes).
- Redraw: Mentally or physically redraw the circuit replacing a block of resistors with their equivalent resistance.
Common mistakes
- Treating a branch as series: Agar do resistors ke beech koi junction aa jaye jahan current split ho sakta hai, toh wo resistors series mein nahi hote.
Recall questions
- If you need an equivalent resistance smaller than any of the available resistors, how should you connect them?
- What is the first step in solving a complex resistor network?
- How do you simplify a series-parallel combination?
Questions & answers
What is the (a) highest, (b) lowest total resistance that can be secured by combinations of four coils of resistance `4 ohm`, `8 ohm`, `12 ohm`, `24 ohm`?
(a) Highest: Series = `48 ohm`. (b) Lowest: Parallel = `2 ohm`.
Approach: Highest = sum them up. Lowest = `1/R = 1/4 + 1/8 + 1/12 + 1/24`.
Two `2Ω` resistors in parallel are connected in series with a `3Ω` resistor. What is the total resistance?
`4 Ω`
Approach: Parallel part: `1Ω`. Total = `1 + 3 = 4Ω`.
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