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.
Understanding Physics: RLC Circuits and Resonance
The inductor and capacitor continually exchange stored energy. An inductor stores energy in its magnetic field when current flows. A capacitor stores energy in its electric field when charge builds on its plates.
During each cycle of the AC supply, energy moves back and forth between these two stores. Near the natural frequency of the circuit, this exchange is timed especially well.
The source then needs to replace mainly the energy lost as heat in the resistor. This is why a small driving voltage can maintain a comparatively large current in a low resistance circuit.
Voltage and current do not always rise and fall together in an RLC circuit. In an inductor, the voltage changes ahead of the current. In a capacitor, the current changes ahead of the voltage.
These timing shifts are called phase differences. Below the resonant frequency, the capacitor has the stronger effect, so the circuit behaves capacitively. Above it, the inductor has the stronger effect, so the circuit behaves inductively.
At resonance, the total voltage from the inductor and capacitor cancels at the circuit terminals, even though each component can have a large voltage across it. This voltage magnification is important in real designs because components must be rated to survive it.
Resistance controls how quickly energy is lost on every cycle. A low resistance circuit loses little energy, so its resonance peak is narrow and tall. It responds strongly over only a small range of frequencies.
A higher resistance circuit has a broader, lower peak. The quality factor describes this tradeoff between sharp frequency selection and energy loss. Bandwidth gives a related practical measure.
It is the range of frequencies around resonance where the response remains fairly strong. In a radio receiver, a narrow bandwidth helps separate one station from nearby stations. If the bandwidth is too narrow, small changes in component values or signal frequency can cause poor reception.
Real components are not ideal. A coil has wire resistance and may lose energy in its magnetic core. A capacitor has leakage and internal resistance.
Connecting wires have small inductance and capacitance of their own. Temperature can change the values of components, which shifts the resonant frequency. Engineers often use adjustable capacitors, carefully wound coils, or electronic feedback to keep a circuit tuned.
When solving school problems, first identify whether the frequency is below, at, or above resonance. Then decide which reactive effect dominates and predict the phase relation before calculating anything.
Keep track of units for inductance, capacitance, frequency, voltage, and current. A sensible physical prediction is a useful check on every calculation.
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.