Visualize the frequency response of a series RLC circuit: adjust R, L, C parameters and observe the magnitude and phase Bode plots in real time
In a series RLC circuit driven by a voltage source, the transfer function H(jω) = V_R / V_in = R / (R + jωL + 1/jωC) describes how the output voltage across R varies with frequency. At the resonant frequency f₀ = 1/(2π√LC), the inductive reactance ωL equals the capacitive reactance 1/(ωC), and they cancel. The impedance is purely resistive (Z = R), so the current and voltage are in phase, and |H| = 1 (0 dB). The Q factor Q = (1/R)√(L/C) measures the sharpness of resonance: high Q means a narrow, peaked response; low Q means a broad, flat response. The -3dB bandwidth is Δf = f₀/Q = R/(2πL), defining the frequency range where |H| ≥ 1/√2.
A Bode plot consists of two graphs sharing the same logarithmic frequency axis. The magnitude plot shows 20·log₁₀|H(f)| in decibels (dB): 0 dB is the unity-gain peak for this passive circuit, and negative dB means attenuation away from resonance. The phase plot shows ∠H(f) in degrees: 0° means in-phase, negative values mean the output lags the input. For the RLC bandpass, the phase transitions from +90° (capacitive, low frequency) through 0° (resonance) to -90° (inductive, high frequency). The steeper the phase transition, the higher the Q. The -3dB points on the magnitude plot define the bandwidth: the frequency range where the response is within 3 dB of its peak.
Radio tuning: LC circuits select a specific station frequency; the Q factor determines selectivity and adjacent channel rejection. Audio crossover networks: RLC filters split audio signals into bass, mid, and treble bands for multi-driver speaker systems. Impedance matching: L-networks use L and C to match antenna impedance to transmission lines at a specific frequency. EMI filters: series LC traps suppress electromagnetic interference at problematic frequencies. Oscillator design: RLC circuits set the oscillation frequency in LC oscillators used in clock generation and RF synthesis. Power electronics: resonant converters use LC tanks for soft-switching, reducing switching losses in high-efficiency power supplies.
Start with the Sharp Resonance preset: a tall, narrow peak appears at the resonant frequency f₀ on the magnitude plot, with a steep phase transition. The stats panel shows high Q and narrow bandwidth. Switch to Broad Response: the peak flattens, Q drops, and bandwidth widens. Try the Audio Filter preset: the resonant frequency moves into the audible range. Drag the R slider to see how resistance controls Q and bandwidth independently of f₀ (which depends only on L and C). Drag L or C to shift the resonant frequency while R controls the peak shape. The vertical dashed line marks f₀; the shaded region shows the -3dB bandwidth. At resonance, the phase passes through 0° — the hallmark of resonance in any second-order system. Hover over either plot to see exact values at any frequency.