AC Circuits

Concept overview

Alternating current circuits are driven by a sinusoidal EMF ε = ε₀ sin(ωt). In steady state, currents are also sinusoidal but possibly phase-shifted from the driving EMF. Root-mean-square values are what meters read: ε_rms = ε₀/√2, I_rms = I₀/√2. Average power dissipated is P = V_rms I_rms cosφ where cosφ is the power factor.

Reactance is the AC analogue of resistance for capacitors and inductors. For an inductor X_L = ωL — increasing with frequency. For a capacitor X_C = 1/(ωC) — decreasing with frequency. Resistors have X = R with no frequency dependence.

In a series RLC circuit, the impedance is Z = √(R² + (X_L − X_C)²), and the current lags the voltage by phase angle tan φ = (X_L − X_C)/R. When X_L = X_C — the resonance condition ω₀ = 1/√(LC) — the impedance is purely resistive, current is maximum, and power factor is unity.

Phasor diagrams give a quick visual solution: draw V_R along the current direction, V_L 90° ahead, V_C 90° behind. The vector sum has magnitude V and direction giving the phase angle. This graphical approach shortens most RLC problems dramatically.

Key formulas & concepts

• RMS values: V_rms = V₀/√2; I_rms = I₀/√2
• Inductive reactance: X_L = ωL (current lags V by 90°)
• Capacitive reactance: X_C = 1/(ωC) (current leads V by 90°)
• Series RLC impedance: Z = √(R² + (X_L − X_C)²)
• Phase angle: tan φ = (X_L − X_C)/R
• Resonance: ω₀ = 1/√(LC); Z_min = R
• Average power: P = V_rms I_rms cosφ
• Q factor (sharpness): Q = ω₀L/R = 1/(ω₀ RC)

Common mistakes to avoid

• Using peak values in the power formula — average power uses RMS values, not peak.
• Adding V_R, V_L and V_C algebraically — they must be added as phasors (vectors with 90° differences).
• Forgetting that at resonance the current is maximum, but the power factor is unity — total power is V_rms × I_rms without the cosφ factor.
• Confusing frequency f with angular frequency ω — formulas usually take ω = 2πf.

JEE Main relevance

JEE Main asks 2 AC problems every paper — usually one series RLC impedance/phase and one power factor or resonance frequency question.

JEE Advanced relevance

Advanced problems combine resonance with the Q factor and bandwidth, or ask about parallel LC circuits at resonance (impedance goes to infinity, not to R). LC oscillation problems that transition into damped oscillations with a resistor are also standard.

Previous-year questions (PYQ)

Series RLC impedance and resonance calculations dominate the PYQ pool. Practise phasor diagrams — they cut problem time in half.