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AC Circuits

Ch. 13Std 12

Easy Overview

Ever wondered why your home appliances run on AC and not DC? Or why a fan makes a humming sound? AC circuits are all about voltages and currents that keep flipping direction — 50 times a second in India. This chapter covers what happens when you put resistors, capacitors, and inductors into an AC circuit. Spoiler: they all behave differently.

AC and Phasors

AC voltage constantly changes like a sine wave — going positive, then negative, then positive again. A phasor is just a rotating arrow that represents this wave. Think of it like a clock hand spinning around — its projection on the x-axis gives you the voltage at any instant. The length of the arrow is the peak voltage, the rotation speed gives you the frequency. Phasors make it way easier to add AC voltages and currents that aren't in sync with each other.

LCR Circuits and Impedance

In an AC circuit, resistance (R), inductive reactance (XL), and capacitive reactance (XC) together form impedance — the total opposition to current. XL = ωL and XC = 1/ωC. These change with frequency. At low frequencies, capacitors block current (high XC). At high frequencies, inductors block current (high XL). The total impedance Z = √(R² + (XL - XC)²). It's like a tug of war — XL and XC oppose each other, while R just does its own thing.

Resonance

Resonance happens when XL equals XC — at that magic frequency, the impedance is minimum (just R), and current is maximum. For an LCR circuit, the resonant frequency is f = 1/(2π√LC). Think of pushing a swing — you get the biggest motion when you push at exactly the natural frequency. Same here — the circuit 'absorbs' the most power at resonance. This is how radio tuners work — you adjust a capacitor to match the resonant frequency to the station you want, filtering out everything else.

Power in AC Circuits

In AC circuits, the voltage and current waves aren't always aligned. There's a phase difference (φ) between them. For a pure resistor, voltage and current are in phase (φ = 0). For a pure inductor, current lags voltage by 90°. For a pure capacitor, current leads voltage by 90°. The real power consumed is P = V_rms × I_rms × cosφ. That cosφ is the power factor. If cosφ = 1 (pure resistive), you're using all the power efficiently. If cosφ = 0 (pure reactive), you're just sloshing energy back and forth, doing no useful work. Low power factor is bad — power companies charge you extra for it.

Key Points

  • •AC voltage: V = V₀ sin(ωt). RMS value = V₀/√2. Same for current.
  • •Inductive reactance XL = ωL (increases with frequency). Capacitive reactance XC = 1/ωC (decreases with frequency).
  • •Impedance Z = √(R² + (XL - XC)²). Phase angle tan φ = (XL - XC)/R.
  • •Resonance in LCR circuit: XL = XC, f = 1/(2π√LC). Current is maximum at resonance.
  • •Power in AC: P = V_rms I_rms cosφ. Only the resistive part consumes real power.
  • •Quality factor Q = ω₀L/R. Higher Q means sharper resonance.

Practice Questions

  • Derive the expression for impedance and phase angle in an LCR series AC circuit.
  • What is resonance in an LCR circuit? Derive the condition for resonance and the resonant frequency.
  • Explain the concept of power factor. How can it be improved?
  • A 100 µF capacitor is connected across a 230 V, 50 Hz AC supply. Find the capacitive reactance and RMS current.