About this model
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
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.