Series LCR Circuit
Z = √(R² + (X_L − X_C)²). Phasor + impedance triangle live.
Key Notes
A series LCR circuit driven by AC has resistor, inductor, and capacitor sharing the SAME current — but with different phase relationships to the source voltage.
Impedance: Z = √(R² + (X_L − X_C)²) — the AC analogue of total resistance.
Peak current: I₀ = V₀/Z. RMS current: I_rms = V_rms/Z.
Phase angle between V and I: φ = arctan((X_L − X_C)/R). Positive φ → V leads I (inductive). Negative → V lags I (capacitive).
Voltage across R is IN PHASE with current; across L leads I by 90°; across C lags I by 90°.
When X_L = X_C (resonance), Z = R (minimum), I is maximum, and circuit behaves purely resistive.
Phasor diagram: V_R along I, V_L at +90° from I, V_C at −90° from I. Net voltage = vector sum.
Average power = V_rms · I_rms · cos φ. The factor cos φ is the POWER FACTOR.
Formulas
Impedance
Resistor combines in QUADRATURE with net reactance.
Reactances
ω = 2πf.
Peak current
Maximum when Z is minimum (resonance).
Phase angle
Sign tells whether circuit is net-inductive or net-capacitive.
Total voltage (phasor)
Pythagorean addition of voltage phasors.
Average power
cos φ = R/Z.
Important Points
V_R, V_L, V_C can individually be MUCH larger than the source V (especially near resonance when V_L = V_C cancel) — but their VECTOR sum equals V.
Power factor cos φ = R/Z. At resonance cos φ = 1; far from resonance, much smaller.
Industrial AC loads are typically inductive (motors, transformers) ⇒ X_L > X_C ⇒ I lags V. Power-factor correction adds capacitors in parallel.
Impedance is FREQUENCY DEPENDENT — same circuit looks different at different frequencies. Filters exploit this.
Vector addition: V_L and V_C are antiparallel (180° apart), so they SUBTRACT in the impedance formula.
If a problem asks 'voltage across L' it means the rms voltage across L alone — which is I_rms × X_L.
Series LCR Circuit notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.