• Tags: LCR, impedance, phase-angle
• Difficulty: Moderate

In a series LCR circuit, the same current I flows through all elements. V_R = IR (in phase with I), V_L = IX_L (leads I by 90), V_C = IX_C (lags I by 90). Since V_L and V_C are anti-phase, the net reactive voltage = |V_L - V_C|. By phasor addition: V = sqrt(V_R^2 + (V_L-V_C)^2). Impedance Z = V/I = sqrt(R^2 + (X_L-X_C)^2). Phase angle: tan(phi) = (X_L-X_C)/R. If X_L > X_C: circuit is inductive, phi > 0, voltage leads current. If X_C > X_L: circuit is capacitive, phi < 0, current leads voltage. Special cases: at resonance (X_L = X_C): Z = R, phi = 0. The impedance triangle is a right triangle with R as base, (X_L-X_C) as height, and Z as hypotenuse. This triangle is the single most useful diagram for solving AC problems.