Build and analyze a current divider

1.2.6 · Hardware › Circuit Analysis Fundamentals

KIYA hai current divider?

Do facts saara kaam karte hain:

KCL (Kirchhoff's Current Law): current in = current out, toh $I_{T} = I_{1} + I_{2}$ .
Parallel branches ek hi voltage $V$ share karti hain.

HOW — do-resistor formula derive karein (scratch se)

Do resistors $R_{1}$ aur $R_{2}$ parallel mein, $I_{T}$ se fed hain.

Step 1 — Same voltage. Kyun? Dono resistors same do nodes ke beech connect hain, isliye unhe identical voltage $V$ milta hai.

$V = I_{1} R_{1} = I_{2} R_{2}$

Step 2 — Equivalent resistance pe Ohm's law se total current. Kyun? Poora bundle ek resistor $R_{∥}$ ki tarah behave karta hai jo $I_{T}$ ko voltage $V$ par carry karta hai.

$V = I_{T} R_{∥}, R_{∥} = \frac{R _{1} R _{2}}{R _{1} + R _{2}}$

Step 3 — $I_{1}$ nikalo. Kyun? $I_{1} = V / R_{1}$ , aur hum $V = I_{T} R_{∥}$ substitute karte hain:

$I_{1} = \frac{V}{R _{1}} = \frac{I _{T} R _{∥}}{R _{1}} = \frac{I _{T}}{R _{1}} \cdot \frac{R _{1} R _{2}}{R _{1} + R _{2}}$

$R_{1}$ cancel ho jaata hai:

Sanity check: dono jodo. $I_{1} + I_{2} = I_{T} \frac{R _{2} + R _{1}}{R _{1} + R _{2}} = I_{T} ✓$

General (N-branch) form conductance use karke

Kyunki $I_{k} = V G_{k}$ aur $I_{T} = V \sum G_{j}$ , divide karo:

Exactly do resistors ke liye yeh wapas boxed formula mein collapse ho jaata hai (try karo — Verify dekho).

Figure — Build and analyze a current divider

Common mistakes (steel-manned)

Recall Feynman: 12-saal ke bacche ko samjhao

Socho paani ek fork pe pahunchta hai jahan do pipes hain. Agar ek pipe wide hai (low resistance) aur ek skinny hai (high resistance), toh zyaada paani wide pipe se bahega. Total paani same rehta hai, bas split ho jaata hai — aur wide, easy pipe bada hissa le jaati hai. Wire mein current exactly yahi karta hai: woh sabse zyaada easiest path se bahta hai.

Active recall

Current divider ke liye branches kaisi honi chahiye — series ya parallel?

Parallel (woh same voltage share karti hain).

Do-resistor current divider formula

I_{1}

ke liye

I_{1} = I_{T} R_{2} / (R_{1} + R_{2})

— doosra resistor upar jaata hai.

Current divider mein "doosra" resistor upar kyun hota hai?

Kyunki

I_{1} = V / R_{1} = I_{T} R_{∥} / R_{1}

, aur

R_{∥}

mein

R_{2}

hai;

R_{1}

cancel ho jaata hai aur

R_{2}

upar reh jaata hai. Current badi resistance se bachta hai.

General N-branch current divider formula

I_{k} = I_{T} G_{k} / \sum_{j} G_{j}

jahan

G = 1/ R

; current conductance ke proportion mein split hota hai.

Parallel network mein current ya voltage divide hota hai?

Current divide hota hai; voltage saari branches mein common hota hai.

Divider mein kaun si branch sabse zyaada current carry karti hai?

Sabse kam resistance (sabse zyaada conductance) wali branch.

Branch currents sahi se sum hoti hain yeh check karo

I_{1} + I_{2} = I_{T} (R_{2} + R_{1}) / (R_{1} + R_{2}) = I_{T}

— KCL satisfy karta hai.

Connections

Kirchhoff's Current Law — woh conservation law jo $I_{T} = \sum I_{k}$ ke peeche hai.
Voltage divider — series dual; formulas contrast karo.
Parallel resistance — $R_{∥}$ derivation mein use hota hai.
Ohm's Law — $I = V / R$ har branch pe apply hota hai.
Conductance — $G = 1/ R$ , current sharing ki natural language.