A series RLC circuit contains a resistor, inductor, and capacitor connected in one path to an AC voltage source. These circuits matter because they model radios, filters, sensors, speakers, and many systems that respond differently to different frequencies. The key idea is impedance, which combines resistance with the frequency-dependent opposition caused by the inductor and capacitor.
At one special frequency, the circuit reaches resonance and the current becomes as large as the resistance allows.
In a series RLC circuit, inductive reactance increases with frequency while capacitive reactance decreases with frequency. Resonance occurs when XL = XC, so their effects cancel and the total impedance is smallest. On a current-versus-frequency graph, this appears as a resonance peak centered at the resonant frequency.
The sharpness of the peak depends on resistance and is described by the quality factor, which helps explain tuning and frequency selection.
Key Facts
- Inductive reactance: XL = 2πfL
- Capacitive reactance: XC = 1/(2πfC)
- Series RLC impedance: Z = sqrt(R^2 + (XL - XC)^2)
- Current amplitude: I = V/Z
- Resonant frequency: f0 = 1/(2πsqrt(LC))
- At resonance in a series RLC circuit, XL = XC, Z = R, and current is maximum.
Vocabulary
- Impedance
- Impedance is the total opposition a circuit gives to AC current, including resistance and reactance.
- Reactance
- Reactance is the frequency-dependent opposition to AC current caused by inductors and capacitors.
- Resonance
- Resonance is the condition in a series RLC circuit when inductive and capacitive reactance are equal and cancel.
- Resonant Frequency
- The resonant frequency is the frequency at which a series RLC circuit has minimum impedance and maximum current.
- Quality Factor
- Quality factor is a measure of how sharp or selective the resonance peak is around the resonant frequency.
Common Mistakes to Avoid
- Adding XL and XC directly in a series RLC impedance calculation is wrong because they oppose each other in phase, so the net reactance is XL - XC.
- Assuming resonance means zero impedance is wrong because the resistor still provides resistance, so the minimum impedance is Z = R.
- Thinking the capacitor blocks all AC current is wrong because capacitive reactance depends on frequency and becomes smaller at higher frequencies.
- Using f0 = 1/sqrt(LC) without the 2π is wrong because that expression gives angular frequency in rad/s only if written as ω0 = 1/sqrt(LC).
Practice Questions
- 1 A series RLC circuit has R = 20 Ω, L = 0.50 H, C = 20 μF, and is driven at f = 60 Hz. Calculate XL, XC, Z, and the current amplitude if the voltage amplitude is 10 V.
- 2 Find the resonant frequency of a series RLC circuit with L = 0.25 H and C = 10 μF.
- 3 A student increases the resistance in a series RLC circuit but leaves L and C unchanged. Explain how the resonance peak changes and whether the resonant frequency changes.