A series RLC circuit you can sweep in your browser
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A resistor, an inductor and a capacitor in series ring at one frequency, and almost every radio, filter and oscillator uses that fact. This page is a working RLC circuit: a signal generator into 250 ohms, 10 mH and 2.5 nF, with a scope on the input and on the point between the inductor and the capacitor.
Press Run and open the Scope tab. The generator is set to 31.8 kHz, right at resonance, and the wave across the capacitor is nearly eight times as tall as the wave going in. Open it in the editor and change the frequency, or sweep it from the Analysis menu, and watch the peak appear and fall away.
The two formulas
The resonant frequency is where the inductor’s reactance and the capacitor’s reactance are equal and cancel: f0 = 1 / (2 pi sqrt(L C)). With L = 10 mH and C = 2.5 nF that is 1 / (2 pi sqrt(0.01 x 2.5e-9)), 31,831 Hz.
How sharp the peak is has a name, Q. For a series RLC it is Q = (1 / R) sqrt(L / C) = (1 / 250) sqrt(0.01 / 2.5e-9) = 8.0. At resonance the voltage across the capacitor is about Q times the input, which is why the scope shows a wave eight times taller than the one that drives it. Nothing is created: the inductor and the capacitor pass energy back and forth, and the resistor is the only part that loses any.
What the sweep shows
Open the circuit in the editor, choose Analysis, then Frequency response, drive gen1 and watch the node between the inductor and the capacitor, from 1 kHz to 1 MHz. Here is what the engine gives.
Well below resonance the line is flat at 0 dB. The capacitor is close to an open circuit, so the whole input appears across it. Near 31.6 kHz the line peaks at 17.9 dB, a ratio of 7.9, a hair under the formula’s 8.0 because the scope’s own 1 megohm input sits across the capacitor. At the same point the phase passes -90 degrees, which is the mark of resonance.
Above resonance the line falls at 40 dB per decade: about -19 dB at 100 kHz and about -60 dB at 1 MHz. That is twice the slope of an RC filter, because two reactive parts are working against the signal instead of one. Taken across the capacitor, a series RLC is a second-order low pass filter.
Change one part at a time
Double R to 500 ohms. The peak falls to 12.0 dB, a ratio of 4.0, exactly half: Q depends on R, and R alone. The resonant frequency does not move.
Halve C to 1.25 nF. The resonant frequency rises by the square root of two, to about 45 kHz, and Q rises too, because sqrt(L / C) grew. The sweep peaks near 44.7 kHz at 20.7 dB.
Change L. It works the same way as C under the square root: four times the inductance halves the resonant frequency.
Measure across the resistor instead, and the same circuit is a band pass filter: the output is largest at resonance and falls off on both sides. Across the inductor, it is a high pass. The scope measures to ground, so reorder the three parts in the editor until the one you want sits at the ground end. One circuit, three filters, depending on where you look.
Series and parallel RLC
In a series RLC the impedance is smallest at resonance, so the current is largest, and the voltages across L and C can each be far bigger than the input. In a parallel RLC it is the other way around: the impedance is largest at resonance and the current drawn from the source is smallest. Radios use a parallel tank to pick one station because of that high impedance at one frequency.
Both have the same f0 formula. Q is written differently: for a parallel circuit it is R sqrt(C / L), so a bigger parallel resistor gives a sharper peak, the reverse of the series case. You can build a parallel version in the editor from the same three parts.
Common mistakes
Mixing units. 10 mH is 0.01 H and 2.5 nF is 2.5e-9 F. Most wrong answers for f0 are off by a factor of 1000 or the square root of it.
Forgetting the source and the probe. A real generator has 50 ohms inside it and a scope input is 1 megohm. Here the generator’s resistance sits before the node being measured and cancels out of the ratio, and the scope trims the peak only slightly, from 8.0 to 7.9. With a smaller inductor and capacitor, or a cheaper scope input, the effect is bigger.
A sweep that is too coarse. A Q of 8 has a narrow peak, and a sweep with a few points per decade can step right over it. Use 40 points a decade or more.
Expecting the peak exactly at f0. Across the capacitor the true peak sits just below f0, at f0 sqrt(1 - 1 / (2 Q squared)). For a Q of 8 that is about 31.7 kHz, too close to tell apart by hand.
Questions
How do I find the resonant frequency of an RLC circuit?
f0 = 1 / (2 pi sqrt(L C)). For 10 mH and 2.5 nF, about 31.8 kHz. The resistor does not change f0; it only sets how sharp the peak is.
What is Q in an RLC circuit?
The quality factor: how sharp the resonance is. For a series RLC, Q = (1 / R) sqrt(L / C). At resonance the voltage across the capacitor is about Q times the input.
Can I see a Bode plot of an RLC circuit online?
Yes. Open the circuit above in the editor, choose Analysis, then Frequency response, and sweep it. It plots gain and phase against frequency, free and with no account.
Is this a SPICE simulator?
The canvas is solved by nodal analysis with real device models, and the editor also runs SPICE netlists. The page on the SPICE simulator says what it runs and what it does not.
Keep going
Build this for real
Open the editor, change a value and watch the number move with it. Nothing to install, and no account needed.