Understand RL transient behavior — Hinglish notes

1.2.8 · Hardware › Circuit Analysis Fundamentals

KYA ho raha hai (setup)

Key inductor law: inductor ki voltage is baat par depend karti hai ki current kitni tezi se change hoti hai: $v_{L} = L \frac{d i}{d t}$

Agar current constant hai ( $d i / d t = 0$ ) → $v_{L} = 0$ → inductor ek plain wire ki tarah dikhta hai (short).
Switch flip ke instant par, inductor current ko continuous rehne par majboor karta hai — $i (0^{+}) = i (0^{-})$ — kyunki instant jump ke liye infinite voltage chahiye hogi.

Current derive KAISE karein (first principles se)

Circuit: battery $V$ , resistor $R$ , inductor $L$ series mein, switch $t = 0$ par close hua.

Step 1 — Kirchhoff's Voltage Law. $V = v_{R} + v_{L} = i R + L \frac{d i}{d t}$ Yeh step kyun? Loop ke around voltage drops ka sum source ke barabar hota hai — energy conservation.

Step 2 — Separable ODE mein rearrange karo. $L \frac{d i}{d t} = V - i R \Rightarrow \frac{d i}{V - i R} = \frac{d t}{L}$ Kyun? Current terms ek side par group karo, time doosri side par, taaki hum integrate kar sakein.

Step 3 — Dono sides ko integrate karo ( $t = 0$ , $i = 0$ se time $t$ , current $i$ tak): $- \frac{1}{R} ln (V - i R)_{0}^{i} = \frac{t}{L}$ $- \frac{1}{R} [ln (V - i R) - ln V] = \frac{t}{L}$ Kyun? $\int \frac{d i}{V - i R} = - \frac{1}{R} ln (V - i R)$ substitution $u = V - i R$ se.

Step 4 — $i$ ke liye solve karo. $ln \frac{V - i R}{V} = - \frac{R}{L} t \Rightarrow V - i R = V e^{- R t / L}$

Discharge (de-energizing): $t = 0$ par source ko short karo initial current $I_{0}$ ke saath. Ab $V = 0$ : $0 = i R + L \frac{d i}{d t} \Rightarrow i (t) = I_{0} e^{- t / τ}$

Figure — Understand RL transient behavior

Time constant KYUN matter karta hai

$t$	$i / I_{final}$ (rising)	remaining (decay)
$1 τ$	63.2%	36.8%
$2 τ$	86.5%	13.5%
$3 τ$	95.0%	5.0%
$5 τ$	99.3%	0.7%

Worked Examples

Common Mistakes

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tum ek bhaari shopping cart dhakka de rahe ho. Jab green light ho jaati hai (switch on), cart ek dam full speed par nahi jaati — usse thoda time lagta hai start hone mein, phir cruise karti hai. Jab tum dhakka dena band karo, woh ekdum nahi rukti — thodi der roll karti hai aur slow hoti hai. Inductor wahi "bhaari pan" hai electric current ke liye: speed up hone mein time lagta hai aur slow down hone mein bhi. Kitna time? Ek number jise $τ = L / R$ kehte hain: bhaari coil ( $L$ ) = slower, zyada friction ( $R$ ) = jaldi settle.

Flashcards

Series RL circuit ka time constant kya hota hai?

τ = L / R

(seconds)

Inductor mein current instantly jump kyun nahi kar sakta?

Kyunki

v_{L} = L d i / d t

; instant jump ke liye infinite voltage chahiye, isliye

i

continuous rehta hai (

i (0^{+}) = i (0^{-})

Energizing RL circuit ke liye rise equation kya hai?

i (t) = \frac{V}{R} (1 - e^{- t / τ})

De-energizing RL circuit ke liye decay equation kya hai?

i (t) = I_{0} e^{- t / τ}

1τ ke baad final value ka kitna fraction reach ho jaata hai (rising)?

63.2%

Kitne τ ke baad transient essentially complete ho jaata hai?

Lagbhag 5τ (>99%).

Switch-on ke turant baad inductor voltage (

t = 0^{+}

Puri source voltage

V

(kyunki

i = 0

, saara drop

L

par hai).

Bada R RL transient ko faster banata hai ya slower?

Faster —

R

denominator mein hai

τ = L / R

mein.

DC ke saath ideal inductor ka steady-state behavior?

Short ki tarah act karta hai (plain wire);

v_{L} = 0

kyunki

d i / d t = 0

Series RL ke liye source V ke saath KVL equation?

V = i R + L d i / d t

Connections

RC Transient Behavior — same math, lekin $τ = R C$ aur charge/current ke roles swap ho jaate hain.
Time Constant — universal exponential-settling parameter.
Inductor Fundamentals — jahan $v_{L} = L d i / d t$ aur energy $\frac{1}{2} L I^{2}$ aate hain.
Kirchhoff's Voltage Law — woh loop equation jahan se humne start kiya.
First-Order Differential Equations — woh ODE technique jo $i (t)$ derive karne ke liye use ki.
Steady-State Analysis — $5 τ$ ke baad circuit kaisa dikhta hai.