An RLC series circuit driven by an alternating voltage source is a fundamental model for understanding frequency-dependent behavior in AC systems. This simulator visualizes the circuit's response by solving the governing differential equation derived from Kirchhoff's voltage law: V_source(t) = V_R + V_L + V_C. The source voltage is V_s = V_0 sin(ωt), where ω = 2πf is the angular frequency. The voltages across the resistor (R), inductor (L), and capacitor (C) are given by V_R = I R, V_L = L dI/dt, and V_C = Q/C, respectively. The core result is the complex impedance, Z = R + j(ωL - 1/(ωC)), whose magnitude |Z| = sqrt(R^2 + (ωL - 1/(ωC))^2) determines the current amplitude: I_0 = V_0 / |Z|. The phase difference φ between the source voltage and the current is φ = arctan((ωL - 1/(ωC))/R). The model simplifies reality by assuming ideal, linear components with constant R, L, and C values, and a perfect sinusoidal source. By interacting with the controls for R, L, C, and f, students directly explore the resonance condition where ωL = 1/(ωC). At this resonant frequency, f_r = 1/(2π√(LC)), |Z| is minimized (equal to R), current is maximized, and the phase φ becomes zero. The simulator displays the resulting |Z|(f) and I(f) curves, the phase plot, and live waveforms for voltage and current, linking the abstract phasor concepts to tangible time-domain signals. This allows learners to internalize how impedance, resonance, and phase shift govern the behavior of filters, tuners, and many electronic systems.
Who it's for: Undergraduate physics and electrical engineering students studying AC circuit theory, resonance, and impedance in the context of second-order linear systems.
Key terms
Impedance
Resonant Frequency
Phase Angle
Reactance
Kirchhoff's Voltage Law
Phasor
Quality Factor (Q)
RLC Circuit
Live graphs
How it works
Series R–L–C driven by V(t) = V₀ sin ωt. Impedance Z = R + j(ωL − 1/ωC). At ω₀ = 1/√(LC) the reactances cancel (X_L = X_C), |Z| = R is minimum, and current is maximum — series resonance. Current lags voltage when the load is net inductive, leads when net capacitive.
Key equations
|Z| = √(R² + (ωL − 1/ωC)²) · I = V₀/|Z| · tan φ = (ωL − 1/ωC) / R
Frequently asked questions
Why does the current reach a maximum at one specific frequency?
The current is maximized when the impedance |Z| is at its minimum. This occurs at the resonant frequency, f_r, where the inductive reactance (ωL) and capacitive reactance (1/ωC) are equal in magnitude but opposite in sign. They cancel each other out in the imaginary part of the impedance, leaving only the resistance R to oppose the current. Thus, the circuit behaves as if it were purely resistive at resonance.
What does a positive or negative phase angle mean for the voltage and current waves?
The phase angle φ is defined as the angle by which the voltage leads the current. A positive φ (common when f > f_r, inductive dominance) means the voltage waveform peaks before the current waveform. A negative φ (when f < f_r, capacitive dominance) means the current peaks before the voltage. At resonance (f = f_r), φ = 0 and the waves are in phase.
How is this circuit used in real-world applications?
RLC series circuits are the basis for band-pass and band-stop filters in radios and communication devices, allowing selection of a specific frequency band. The sharpness of the resonance peak, quantified by the Quality Factor (Q = ω_rL/R), determines the filter's selectivity. They are also fundamental in modeling antenna tuning circuits and the response of resonant sensors.
What are the limitations of this ideal model?
This model assumes perfect components. Real inductors have internal resistance (beyond the series R), capacitors have leakage, and all components have frequency-dependent parasitic effects at very high frequencies. The source is assumed to be ideal with zero internal impedance. These simplifications allow clarity in learning core principles before addressing non-ideal behavior.