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CR · Resonance

Series RLC Resonance

Inductors and capacitors respond to frequency in opposite ways. At the one frequency where their responses cancel exactly, the circuit opens up the most; this unit shows how to find and work with that resonant frequency.

The one point where the reactances cancel

The graph's horizontal axis is frequency on a logarithmic scale, and the vertical axis is current relative to its maximum. Raise the frequency from low to high: at low frequencies the capacitor blocks the way, at high frequencies the inductor does, and the current stays small. Find the resonant frequency ω_0, where the two reactances cancel exactly and the current rises highest.

Frequency ω / ω_0ω/ω_0 = 0.32
Current (relative to peak)
I = 9% · |Z| = 11.43 R
Far from resonance

Inductor and capacitor are opposites

The inductive reactance X_L = ωL grows in proportion to frequency, while the capacitive reactance X_C = 1/ωC shrinks in inverse proportion. On top of that, the two point in opposite directions in the complex plane (+j and −j). So in series the two reactances do not add but subtract, and the net reactance is X_L − X_C.

The resonant frequency ω_0 = 1/√(LC)

The resonance point is the frequency where the net reactance becomes 0, that is, where X_L = X_C. Solving ωL = 1/ωC gives ω_0 = 1/√(LC). At this frequency the imaginary part of the impedance vanishes, leaving only the resistance R, so the impedance is smallest; apply the same voltage and the current V/R flows at its maximum. The circuit opens up most widely at that frequency.

Q sets the sharpness of the peak

The smaller the resistance, the taller and narrower the resonance peak, and its sharpness is measured by the quality factor Q = ω_0L/R. A large Q gives a sharp selectivity that passes only a narrow band around the resonant frequency, while a small Q passes a broad range. Picking out the station you want on a radio is exactly this selectivity at work. This plot fixes Q at 4 and scales the peak height to 1.

ObserveXL = XC
At resonance the two reactances are equal.
Chooseω0 = ?
It comes from solving ωL = 1/ωC.
Fill inω = ω0 → Z = ?
The reactances cancel, leaving only resistance.
On your ownQ = ?
Smaller resistance means larger Q.

Back to the first screen

At low frequencies the capacitor blocked the way and at high frequencies the inductor did, so the current was small. At ω_0 = 1/√(LC), where the two reactances become exactly equal, they cancelled, the impedance fell to the resistance R, and the current shot up into a sharp peak. What set the height and width of that peak was the quality factor Q.

Resonance is the condition in a series RLC circuit where the inductive and capacitive reactances become equal and cancel. At the resonant frequency ω0 = 1/√(LC) the impedance falls to its minimum, the resistance R, and the current is largest. The sharpness of the peak is set by the quality factor Q = ω0L/R; the larger Q is, the more selectively the circuit passes only a narrow band.