Series RLC Resonance

Worked Example

Consider an RLC circuit driven by a 1 V RMS sinusoidal source with parameters: Resistance R = 2 Ω, Inductance L = 1 H, Capacitance C = 4 F, and Angular frequency ω = 1 rad/s. We calculate the steady-state response as follows:

1. Calculate the reactance X = ωL - 1/(ωC) = (1)(1) - 1/((1)(4)) = 1 - 0.25 = 0.75 Ω.
2. Calculate the impedance magnitude |Z| = √(R² + X²) = √(2² + 0.75²) = √(4 + 0.5625) = √(4.5625) ≈ 2.1360009363294 Ω.
3. Calculate the RMS current I = 1 / |Z| = 1 / 2.1360009363294 ≈ 0.46816458878452 A.
4. Calculate the phase angle φ = -atan2(X, R) = -atan2(0.75, 2) ≈ -20.556045219583°. The negative sign means the current lags the source voltage.
5. Calculate individual voltages: V_R = I · R ≈ 0.93632917756904 V, V_L = I · ωL ≈ 0.46816458878452 V, V_C = I / (ωC) ≈ 0.11704114719613 V.
6. Verify that V_R² + (V_L - V_C)² = 1.0 V², matching the source voltage squared, with real power P = I²R ≈ 0.43835616438356 W.

Experiment

Open the Series RLC Resonance module to observe these principles in action:

1. Set Resistance R to 1, Inductance L to 1, and Capacitance C to 1.
2. Set the Angular frequency omega to 1. Observe that the reactance is 0 and the current is 1 A (dissipative resonance), with 0 degrees phase difference.
3. Change Angular frequency omega to 1/2. The circuit becomes capacitive (current leads voltage), and the reactance drops to -1.5 Ω. The phase shifts to about 56.31 degrees.
4. Change Resistance R to 0 and Angular frequency omega back to 1. Notice that the status changes to “No finite resonant phasor” and values become unavailable because there is no resistance to dissipate energy at resonance.

Limitations

The steady-state phasor analysis assumes that the driving source has been operating in sinusoidal steady state, so initial switching transients have decayed. The model assumes ideal components with no parasitic resistance in the inductor or capacitor. When resistance is set to zero precisely at resonance, the model reports “No finite resonant phasor” because an ideal lossless LC circuit driven at resonance has zero impedance and no finite forced sinusoidal phasor exists.

References

The following reference supports the AC impedance, reactance cancellation, and phasor relationships used in this module:

• Section 15.3, RLC Series Circuits with AC — OpenStax University Physics Volume 2.