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f₀ resonance
Series RLC driven oscillator
A series circuit of a resistor (R), inductor (L) and capacitor (C) is driven by an alternating voltage source. As you sweep the drive frequency, the circuit's response peaks sharply at its natural resonant frequency — exactly the effect that lets a radio pick out one station from many.
Z = √(R² + (ωL − 1/(ωC))²)
ω₀ = 1/√(LC), f₀ = 1/(2π√LC)
Q = ω₀L / R
I = V / Z, φ = atan((ωL − 1/(ωC)) / R)
At resonance the voltage across the inductor (and capacitor) can be Q times larger than the source voltage — a high-Q circuit can produce hundreds of volts from a few volts of drive, which is how crystal radios extract a faint signal.
What is electrical resonance in an RLC circuit?
Resonance occurs when the inductive reactance and capacitive reactance cancel exactly, so the circuit's impedance equals just the resistance R. The current driven by an AC source then reaches its maximum value.
What is the resonant frequency formula?
The resonant angular frequency is ω₀ = 1/√(LC), giving a resonant frequency f₀ = 1/(2π√LC). It depends only on the inductance L and capacitance C, not on the resistance.
How is the impedance of a series RLC circuit calculated?
Impedance is Z = √(R² + (ωL − 1/(ωC))²). The term ωL is the inductive reactance and 1/(ωC) is the capacitive reactance; their difference is the net reactance.
The quality factor Q = ω₀·L/R measures how sharp the resonance peak is. A high Q means low resistance, a narrow bandwidth and a tall, selective resonance curve.
Because the inductive and capacitive reactances cancel, the impedance drops to its minimum value R. Since current amplitude I = V/Z, minimum impedance produces maximum current.
At resonance the voltage and current are in phase (phase angle φ = 0). Below resonance the circuit is capacitive (current leads); above resonance it is inductive (current lags).
Higher resistance lowers and broadens the resonance peak (low Q), while lower resistance produces a taller, sharper peak (high Q). Resistance sets the bandwidth Δf = f₀/Q.
A tuned RLC circuit responds strongly only to signals near its resonant frequency. Varying C (or L) shifts f₀ so the receiver selects one station while rejecting others.
The bandwidth is the frequency range between the half-power (−3 dB) points where current falls to 1/√2 of its peak. It equals Δf = f₀/Q = R/(2πL).
A phasor diagram represents the AC voltages across R, L and C as rotating vectors. Adding the phasors shows the total voltage and the phase difference between voltage and current.