RLC Circuits

Add an inductor to the RC circuit and you get resonance — the ability to select or reject a specific frequency. RLC circuits are the basis of radio tuning, bandpass filters, notch filters, and any application needing frequency selectivity. The interplay between capacitor (storing energy in an electric field) and inductor (storing energy in a magnetic field) creates oscillatory behaviour that neither component can produce alone, with the resistor determining how much energy is lost per cycle.

Resonant Frequency

At the resonant frequency, inductive reactance and capacitive reactance are equal and opposite — they cancel. What's left is purely resistive:

f₀ = 1 / (2π√(LC))

At resonance: X_L = X_C
  2πf₀L = 1/(2πf₀C)
LCf₀Typical application
100 µH100 pF1.59 MHzAM broadcast band tuning
10 µH100 pF5.03 MHzShortwave receiver
1 µH47 pF23.2 MHzCB radio, 27 MHz ISM
1 mH1 µF5.03 kHzAudio notch filter
10 mH10 µF503 HzPower line filter
100 mH100 µF50.3 HzMains frequency trap

Series RLC Circuit

Series RLC Circuit Schematic diagram showing a resistor, inductor, and capacitor connected in series. Vin R L C GND (or Vout tap)
A basic series RLC circuit. Different filter responses are achieved depending on which component the output voltage is measured across.

Impedance at any frequency:

Z = R + j(X_L − X_C)
Z = R + j(2πfL − 1/(2πfC))

Magnitude: |Z| = √(R² + (X_L − X_C)²)
Phase:     θ = arctan((X_L − X_C) / R)

At f < f₀:  X_C > X_L  → circuit is capacitive, current leads voltage
At f = f₀:  X_L = X_C  → Z = R (minimum), current is maximum
At f > f₀:  X_L > X_C  → circuit is inductive, current lags voltage

Output Taken Across Different Elements

Output acrossFilter responseUse
Resistor RBandpass — maximum at f₀Frequency selection, radio IF filter
Capacitor CLow-pass below resonance, peaks at f₀Voltage magnification at resonance
Inductor LHigh-pass above resonance, peaks at f₀Voltage magnification at resonance
L + C in seriesBandpass — sharp peak at f₀Narrow-band filter

Parallel RLC Circuit

Parallel RLC Circuit Schematic diagram showing a resistor, inductor, and capacitor connected in parallel between Vin and Vout. Vin Vout R L C
A parallel RLC circuit, often called a tank circuit. At resonance, the circulating current between the inductor and capacitor can be much larger than the source current.

At resonance, the parallel LC "tank circuit" has maximum impedance. Current from the source is minimum; circulating current within the tank can be large. This is the opposite of the series case:

Series RLC at resonance: minimum impedance (current maximum)
Parallel RLC at resonance: maximum impedance (current minimum, voltage maximum)

Quality Factor (Q)

Q measures the sharpness of the resonance — how well the circuit discriminates between f₀ and nearby frequencies:

Series RLC:   Q = (1/R) × √(L/C) = X_L/R = X_C/R = f₀/BW
Parallel RLC: Q = R × √(C/L) = R/X_L = R/X_C

Bandwidth:    BW = f₀ / Q
Q factorBandwidth at f₀=1MHzSelectivityApplication
11 MHz (very wide)PoorBroadband matching
10100 kHzModerateAudio bandpass filters
5020 kHzGoodAM radio IF filter (455 kHz IF)
10010 kHzHighShortwave receiver front end
10,000+<100 HzVery highQuartz crystal resonator

Voltage Magnification at Resonance

In a series RLC at resonance, the voltage across L or C individually can be Q times the supply voltage:

V_L = V_C = Q × V_in   (at resonance)

Example: Q=50, V_in=1V
  Voltage across capacitor: 50V
  Voltage across inductor:  50V
  (they're equal and opposite — cancel in series, but each is large alone)

This is why high-Q RF circuits need high-voltage-rated capacitors
even when the supply voltage is low.

Damping

The resistance determines the transient behaviour — what happens when the circuit is disturbed from equilibrium:

Damping typeConditionTransient responseUse case
UnderdampedQ > 0.5 (R small)Oscillates, rings. Exponentially decaying sine wave.High-Q filters, oscillator tanks — ringing is wanted
Critically dampedQ = 0.5 exactlyFastest return to steady state with no overshootControl systems, step response optimisation
OverdampedQ < 0.5 (R large)Slow exponential return, no oscillationDamping unwanted resonances in power supplies

Practical Circuits

AM Radio Tank Circuit

AM Radio Tank Circuit An antenna connected to a ferrite rod inductor in series, with a variable tuning capacitor connected in parallel to ground, and an output to the detector. Antenna Ferrite Rod Inductor To detector ← tune to station
A practical AM radio tuning circuit. The ferrite rod acts as both the antenna and the inductor, while a variable capacitor sets the resonant frequency.
L ≈ 200–500 µH (ferrite rod coil)
C = 10–350 pF variable (tuning capacitor)
f₀ range covers ~540 kHz – 1700 kHz (AM band)

At 1 MHz with L=250µH: C = 1/(4π²f₀²L) = 101 pF

60Hz Notch Filter for Audio

Series LC in shunt path (Twin-T alternative):
60Hz LC Notch Filter A signal path with a series LC branch connected to ground acting as a shunt notch filter. Signal Signal out L C ← series LC to GND
A passive LC notch filter. At the resonant frequency, the series LC circuit has minimum impedance and shorts the unwanted frequency to ground.
L and C chosen so f₀ = 60 Hz:
With L=1H (large): C = 1/(4π²×60²×1) ≈ 7 µF

In practice: use active twin-T notch (op-amp) for audio — achieves higher Q
and avoids the large, expensive, lossy 1H inductor.

Output Filter for Switching Power Supply

Buck converter output filter:
  L and C form a 2nd-order low-pass filter
  Switching noise (100kHz+) is attenuated; DC and slow changes pass

f_corner = 1/(2π√LC) should be << switching frequency
f_corner should be >> desired output bandwidth

Typical: f_switch=200kHz, f_corner=5-20kHz
L=10µH, C=100µF: f₀=5.03kHz → attenuates 200kHz by ~32dB

References