LCR Resonance
I peaks at f₀ = 1/(2π√LC). Q-factor and bandwidth Δf = f₀/Q.
Key Notes
Series LCR resonance: at f₀ = 1/(2π√(LC)), the inductive and capacitive reactances cancel and impedance is minimum (= R).
Current is MAXIMUM at resonance: I_max = V_rms/R, regardless of L or C separately.
At resonance, V and I are IN PHASE — circuit looks purely resistive even though L and C are present.
Quality factor Q = (1/R)√(L/C) = ω₀L/R = 1/(ω₀RC). Higher Q ⇒ sharper resonance, narrower bandwidth.
Bandwidth Δf = f₀/Q = R/(2πL). Wide R (resistance) ⇒ broad resonance peak. Low R ⇒ sharp, selective.
Resonance is the principle of RADIO TUNING — varying C tunes Q-circuit to the broadcast frequency.
V across L and V across C can each be Q × V_source — much larger than the source. This is voltage MAGNIFICATION at resonance.
Parallel (anti-)resonance behaves OPPOSITELY: impedance is MAXIMUM, current minimum (for ideal components).
Formulas
Resonance frequency
Series LCR with X_L = X_C.
Maximum current
At f = f₀; circuit is purely resistive.
Quality factor
All three forms equivalent.
Bandwidth
Half-power width (between I_max/√2 points).
Voltage magnification
At resonance — can be very large.
Important Points
Resonance frequency depends only on L and C — NOT on R. R determines the SHARPNESS, not the location.
Q quantifies how 'selective' the resonance is. Q = 100 means the half-power bandwidth is 1% of f₀ — very narrow.
A low-Q circuit accepts a broad range of frequencies; a high-Q circuit accepts only a narrow band.
At resonance, source supplies energy ONLY to dissipate in R. Energy bounces between L and C with no source involvement otherwise.
V_L and V_C can EXCEED source voltage by factor Q — design must account for component voltage ratings.
Don't confuse: at resonance, impedance is MINIMUM in series LCR, MAXIMUM in parallel LCR (for ideal L, C).
LCR Resonance 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.