CR · Resonance

Series RLC Resonance

An inductor and a capacitor react to frequency in exactly opposite ways. At one frequency their reactions cancel precisely and the circuit opens up the widest. Learn to find and handle that resonant frequency.

The one point where the reactances cancel

Sweep the frequency from low to high and watch the current. Low down the capacitor blocks it, up high the inductor blocks it, so the current is small. Find the resonant frequency ω_0 where the two cancel exactly and the current peaks.

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

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

The resonant frequency ω_0 = 1/√(LC)

The frequency where the net reactance is zero — where X_L = X_C — is the resonant point. Solving ωL = 1/ωC gives ω_0 = 1/√(LC). At this frequency the impedance loses its imaginary part and only the pure resistance R is left, so it is at its smallest. With the same voltage applied the current V/R is at its maximum, and the circuit opens widest at that frequency.

Q sets the sharpness of the peak

The smaller the resistance, the higher and narrower the resonant peak. That 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; a small Q passes a wide range. Tuning a radio is exactly putting this resonant selectivity to use.

ObserveX_L = X_C

At resonance the two reactances are equal.

Chooseω₀ = ?

It comes from solving ωL = 1/ωC.

Fill inω = ω₀ → Z = ?

The reactance cancels, leaving pure R.

On your ownQ = ?

Smaller resistance means larger Q.

Back to the first screen

At low frequency the capacitor blocked the path, at high frequency the inductor did, so the current was small. At ω_0 = 1/√(LC), where the two reactances are exactly equal, they cancelled, the impedance fell to pure R, and the current rose in a sharp peak. The height and width of that peak were held by the quality factor Q.

Resonance in a series RLC is when the inductive and capacitive reactances become equal and cancel. At the resonant frequency ω₀ = 1/√(LC) the impedance is at its minimum, pure R, and the current is maximal. The sharpness of the peak is set by the quality factor Q = ω_0L/R: the larger Q, the greater the selectivity that passes only a narrow band.