Analyze simple AC circuits with reactance — Hinglish notes

1.2.14 · Hardware › Circuit Analysis Fundamentals

HUM kya analyze kar rahe hain?

Complex numbers kyun? Kyunki AC quantities mein dono hote hain — ek size (amplitude) aur ek timing (phase). Ek akela real number dono hold nahi kar sakta; ek complex number kar sakta hai. $Z$ ka angle hi phase shift hai.

HOW reactance aata hai — first principles se derive karo

Hum har component ko $v (t) = V_{m} cos (ω t)$ se drive karte hain aur poochte hain "kaunsa current flow karta hai?"

Inductor — WHY $X_{L} = ω L$

Inductor ka defining law: $v_{L} = L \frac{d i}{d t}$

Yeh step kyun? Yeh physics hai: inductor ke across voltage proportional hota hai is baat se ki current kitni tezi se change ho rahi hai. Current ke liye integrate karke solve karo: $i (t) = \frac{1}{L} \int V_{m} cos (ω t) d t = \frac{V _{m}}{ω L} sin (ω t)$

Yeh step kyun? $cos$ ko integrate karne par $\frac{1}{ω} sin$ milta hai, aur $\frac{1}{L}$ rearrange karne se aaya.

Ab peak current hai $I_{m} = \frac{V _{m}}{ω L}$ . Ohm's law $I_{m} = V_{m} / X$ se compare karo:

Yeh $f$ ke saath kyun badhta hai? Zyada fast changes → bada $d i / d t$ → zyada opposing voltage chahiye → kam current nikalta hai. Inductors high frequencies ko block karte hain.

Capacitor — WHY $X_{C} = 1/ (ω C)$

Capacitor law: $i_{C} = C \frac{d v}{d t}$

Yeh step kyun? Current plates par charge ka flow hai; har second mein zyada voltage swing = zyada current. $i (t) = C \frac{d}{d t} (V_{m} cos ω t) = - ω C V_{m} sin (ω t) = ω C V_{m} cos (ω t + 90°)$

Peak current $I_{m} = ω C V_{m}$ , toh $V_{m} / I_{m} = 1/ (ω C)$ :

Yeh $f$ ke saath kyun ghatta hai? Fast wiggling voltage charge/discharge karte rehna → bahut current flow hota hai → bahut kam opposition. Capacitors high frequencies pass karte hain.

Impedance magnitude mein combine karna

$R$ , $L$ , $C$ series mein ke liye, impedances jodo (sabka same current hai): $Z = R + j (X_{L} - X_{C})$ Magnitude (Pythagoras, kyunki real aur imaginary parts perpendicular hain): $∣ Z ∣ = R^{2} + (X_{L} - X_{C})^{2}, ϕ = tan^{- 1} \frac{X _{L} - X _{C}}{R}$

Pythagoras kyun? $R$ (in-phase) aur $X$ (90° out-of-phase) do axes ki tarah hain jo right angles par hain.

Figure — Analyze simple AC circuits with reactance

Worked examples

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek bachche ko swing par aage-peeche dhakka de rahe ho. Ek resistor chipchipey keechad ki tarah hai — yeh bas tumhe slow karta hai aur tumhari energy ko heat mein badal deta hai. Ek inductor ek bhaari swing ki tarah hai jo direction change karna pasand nahi karta; jitna tezi se tum use wiggle karne ki koshish karo, utna zyada yeh fight karta hai — toh fast wiggles (high frequency) block ho jaate hain. Ek capacitor ek springy trampoline hai; dheemi gentle pushes use barely hilaati hain (low frequency block karta hai) lekin fast chhote bounces aasani se flow karte hain. Reactance bas "springy ya heavy cheez kitna push back karti hai" hai, aur yeh change hoti hai is baat ke saath ki tum kitni tezi se wiggle karte ho. Jab tum pushback (X) aur keechad (R) ko mix karte ho, tum unhe simply add nahi kar sakte kyunki yeh tumhe alag moments par hit karte hain — toh tum ek rectangle ka diagonal use karte ho (Pythagoras).

Active-recall flashcards

Reactance kya hai aur uski unit kya hai?

Frequency-dependent opposition to AC from L or C; ohms

Ω

mein measure hoti hai; energy store karti hai, dissipate nahi karti.

Inductive reactance ka formula?

X_{L} = 2 π f L = ω L

Capacitive reactance ka formula?

X_{C} = \frac{1}{2 π f C} = \frac{1}{ω C}

Inductor mein, kya current voltage se lead karta hai ya lag?

Current voltage se 90° lag karta hai (ELI).

Capacitor mein, kya current voltage se lead karta hai ya lag?

Current voltage se 90° lead karta hai (ICE).

X_{L}

frequency badhne par kaise change hota hai?

Proportionally badhta hai (high freq block karta hai).

X_{C}

frequency badhne par kaise change hota hai?

Inversely ghatta hai (high freq pass karta hai).

Series RLC impedance ki magnitude?

∣ Z ∣ = R^{2} + (X_{L} - X_{C})^{2}

Impedance ka phase angle?

ϕ = tan^{- 1} \frac{X _{L} - X _{C}}{R}

R aur X ko arithmetically kyun nahi jod sakte?

Yeh 90° out of phase hain; Pythagoras se combine karo.

Capacitor ki complex impedance?

Z_{C} = 1/ (j ω C) = - j / (ω C)

X_{L}

v = L d i / d t

se kyun derive karte hain?

Ek cosine drive ko integrate karne par

I_{m} = V_{m} / (ω L)

milta hai, toh ratio

V_{m} / I_{m} = ω L

hai.

Connections

Ohm's Law — $I = V / Z$ generalizes $I = V / R$ .
Complex Numbers and Phasors — kaise $j$ phase encode karta hai.
Capacitors aur Inductors — woh components jo $X$ produce karte hain.
Resonance in RLC Circuits — jab $X_{L} = X_{C}$ , reactance cancel ho jaati hai.
RMS and Peak Values — voltage/current amplitude conventions.
Power in AC Circuits — power factor $cos ϕ$ isi $ϕ$ se.