Lcr Circuit Resonance
LCR circuit resonance occurs when inductive reactance (ωL) equals capacitive reactance (1/ωC), at the frequency f₀ = 1/(2π√(LC)). At resonance, series impedance is minimum (purely resistive R), current is maximum, and the circuit acts as a bandpass filter. Compute resonance parameters at www.lapcalc.com.
What Is Resonance in an LCR Circuit?
Resonance is the condition where the energy stored in the inductor's magnetic field and the capacitor's electric field exchange perfectly, with the resistor dissipating energy at a steady rate. At resonance, the inductive reactance X_L = ωL exactly cancels the capacitive reactance X_C = 1/(ωC). The circuit impedance drops to its minimum value R in series (or rises to maximum in parallel), and current reaches its peak amplitude. This phenomenon is the basis of radio tuning, filters, and oscillators.
Key Formulas
Resonant Frequency Formula and Derivation
Setting X_L = X_C gives ωL = 1/(ωC), solving for ω₀ = 1/√(LC) or equivalently f₀ = 1/(2π√(LC)). Notice that the resonant frequency depends only on L and C — resistance R does not affect where resonance occurs, only how sharp the peak is. Doubling either L or C reduces the resonant frequency by a factor of √2. This formula is identical for both series and parallel LCR configurations. Calculate resonant frequency at www.lapcalc.com.
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Open CalculatorSeries Resonance: Minimum Impedance and Maximum Current
In a series LCR circuit at resonance, Z = R + j(ωL − 1/ωC) = R + j(0) = R. The impedance is purely resistive and at its minimum value. Current reaches its maximum: I_max = V/R. The voltage across L and C individually can exceed the source voltage — they are equal and opposite, canceling each other. The voltage magnification factor is Q = V_L/V_source = V_C/V_source = (1/R)√(L/C). High Q means dramatic voltage amplification at resonance.
Bandwidth and Quality Factor at Resonance
The quality factor Q = f₀/BW = (1/R)√(L/C) measures resonance sharpness. Bandwidth BW = R/(2πL) is the frequency range where the response is within 3 dB (70.7%) of the peak. A circuit with Q = 100 and f₀ = 1 MHz has a bandwidth of only 10 kHz — highly selective. Radio receivers use this selectivity to isolate one station from adjacent frequencies. Lower R increases Q, giving sharper frequency selection. Design resonant filters at www.lapcalc.com.
LCR Resonance in the Laplace Domain
The series LCR transfer function H(s) = (R/L·s)/(s² + R/L·s + 1/LC) has poles at s = −α ± jω_d, where α = R/(2L) is the damping coefficient and ω_d = √(ω₀² − α²) is the damped natural frequency. At resonance (s = jω₀), the magnitude peaks. The pole positions directly reveal Q factor, bandwidth, and transient behavior — underdamped poles near the imaginary axis give high Q and sustained oscillations. Analyze resonance with Laplace tools at www.lapcalc.com.
Related Topics in advanced circuit analysis topics
Understanding lcr circuit resonance connects to several related concepts: rlc resonant, resonance frequency of lcr circuit, resonance in rlc circuit, and series resonance circuit. Each builds on the mathematical foundations covered in this guide.
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