# Series RLC - how does resonance amplify?

A series RLC circuit has a resonant frequency f0. If I apply a sinusoid at frequency f0 to this series RLC circuit, the output of the RLC should be a oscillation at frequency f0 with an amplitude higher than the sinusoidal input amplitude.

However, a property of a series RLC circuit is that, at resonance, the RLC appears as purely resistive, the reactive components cancel.

So, how is it possible that the output of the RLC can output a higher amplitude than the input sinusoid if the circuit appears purely resistive?

simulate this circuit – Schematic created using CircuitLab

• if you put it that way, it sounds like a good question 😉 I am not yet sure if it really is, or if that wording merely leads to some otherwise very obvious oversight. Jun 11 at 17:39
• Why shouldn't it amplify? Other passive circuits amplify - transformer for example.
– Chu
Jun 11 at 17:50
• @Chu At resonance, the reactive components of the inductor and capacitor cancel each other out. If I subject an RLC circuit to a sinusoidal signal with frequency equal to the resonant frequency of the RLC, the input impedance seen by that source is purely resistive. So my question is : If the input impedance is purely resistive at the resonant frequency, how on earth can it amplify? Jun 11 at 17:57
• Depends where you take the output - draw your circuit to remove ambiguity. Jun 11 at 18:10
• @Andyaka just added Jun 11 at 18:15

Suppose a high-quality LC series circuit is driven by a function generator having internal 50 ohm resistance:

simulate this circuit – Schematic created using CircuitLab

At resonance (1000 Hz) capacitive reactance cancels inductive reactance so that the function generator sees a load very near zero ohms. If function generator amplitude was 1V, then only R1 limits current to 20 mA. This current is in phase with V1.

But this 20 mA must flow through the inductor, whose reactance is 159.153 ohms. So a voltage appears across the inductor whose amplitude is 3.183 volts, over three times higher than the open-loop function generator's voltage. Seems like you're getting some kind of gain, but you should recognize that current and voltage at L1 have a 90-degree phase relationship, so the only real power is dissipated in R1.

The essence of the series LC is that they exchange energy with each other. If you don't allow yourself to "look inside the box" then the series LC looks like a low impedance (zero, if ideal) at their resonant frequency. However, if you do look inside the box (the middle node where you pick off your choice to examine in your schematic) then you can see that this particular interior node is "dancing around" a lot, but where the voltages across the inductor and the capacitor are roughly equal and oppositely arranged to each other so that the voltage sum across the two appears to be close to zero.

What this kind of reminds me of is an old "thread & button spinner" toy I have played with from time to time. You can see it operate on this youtube video (I've specified it to start about where you can see it working well.) Only a little energy is being supplied by the slight pulling needed by the operator's hands every cycle, while the button rotates so fast you can hear the wind whistling by. The motion seen from both ends of the toy is modest. But the motion in the center is wild and fast!

Let's set this up in a simple LTspice schematic:

I'll explain the equations shown there and then get to the results.

The KCL is a little bit annoying, at first:

\begin{align*} \frac1{L_1}\int V_{_\text{OUT}}\:\text{d}t + C_1\frac{\text{d}}{\text{d}t} V_{_\text{OUT}} &= \frac1{L_1}\int V_{_\text{GND}}\:\text{d}t \\\\ \frac{V_{_\text{GND}}}{R_1} + \frac1{L_1}\int V_{_\text{GND}}\:\text{d}t &= \frac{V_{_\text{IN}}}{R_1} + \frac1{L_1}\int V_{_\text{OUT}}\:\text{d}t \end{align*}

But if I take derivatives, solve and substitute a bit, and finally move things around I will wind up with:

$$\frac{\text{d}^3}{\text{d}t^3}V_{_\text{OUT}}+\frac{R}{L}\frac{\text{d}^2}{\text{d}t^2}V_{_\text{OUT}}+\frac1{LC}\frac{\text{d}}{\text{d}t}V_{_\text{OUT}}=\frac1{LC}\frac{\text{d}}{\text{d}t}V_{_\text{IN}}$$

By substitution of $$\Z=V_{_\text{OUT}}^{\quad'}=\frac{\text{d}}{\text{d}t}V_{_\text{OUT}}\$$ I find that the characteristic equation is $$\Z^{''}+\frac{R}{L}Z^{'}+\frac1{LC}Z\$$ with $$\\omega_{_0}=\frac1{\sqrt{LC}}\$$ and $$\\zeta=\frac{R}2\sqrt{\frac{C}{L}}\$$. Knowing that $$\Q=\frac1{2\zeta}\$$ I can find that if I know $$\f_{_0}\$$ and a desired $$\Q\$$, I can solve for the rest as shown in the above schematic.

I've assigned $$\\tau_{_0}=\frac1{\omega_{_0}}\$$. So $$\L=\tau_{_0}\cdot Q\cdot R\$$ and $$\C=\frac{\tau_{_0}}{Q\,\cdot\, R}\$$.

If I assign $$\Q=1\$$ then this is what I get:

Note that $$\V_{_\text{GND}}\$$ stays very close to ground. It doesn't move around that much. But now also note that $$\V_{_\text{OUT}}\$$ (the green trace, which is the capacitor voltage) and that $$\V_{_\text{GND}}-V_{_\text{OUT}}\$$ (the bluish trace, which is the inductor voltage) are moving almost exactly oppositely to each other and, broadly speaking, sum up close to zero all the time. Finally, note that all the magnitudes, including $$\V_{_\text{IN}}\$$, are the same.

Let's change to $$\Q=2\$$:

Note that $$\V_{_\text{GND}}\$$, while still close to ground, is swinging around a little bit more. Note that the peaks for the capacitor and inductor voltages are now twice as large as the input voltage peaks and still act in a way that their sum remains close to zero.

Let's try $$\Q=5\$$:

And now you can see that the peaks for the capacitor and inductor voltages are now five times as large as the input voltage peaks and yet still always sum to near zero. ($$\V_{_\text{GND}}\$$ has somewhat yet-larger swings.)

The key things to notice are that the voltage drop across the resistor remains very close to the source voltage and that the voltage drop across the LC pair remains very close to zero. What's happening "inside the box" is that energy is being transferred back and forth between the inductor and capacitor and that the rate at which the supply voltage itself is changing is just exactly what is needed so that the summed voltage across the inductor and capacitor remains close to zero throughout a cycle. If the source frequency were to be any higher, or any lower, then the timing of the exchange of energy between the inductor and capacitor would no longer match up so well and the effect would be lost.

The L and C cancel one another because the voltages across them are 180 degrees out of phase. The voltages don't both go to 0V.

The key here is to look at the waveforms or the phasor diagram.

Image source: Waveforms for Series AC Circuits

In an ac circuit, current (red) I is in phase with the voltage drop across the resistor (blue) $$\V_R\$$. I lags the voltage drop across the ideal inductor (brown) $$\V_L\$$. I leads the voltage drop across the capacitor (purple) $$\V_C\$$. These are shown in the phasor diagram on the left. The $$\V_L\$$ phasor is the same magnitude and opposite in polarity to the $$\V_C\$$ phasor.

At resonance, they cancel out and the circuit becomes resistive. Similarily, the purple waveform added to the brown waveform will be 0.

Power waveform is shown in green. At resonance, all the power provided by source goes to the resistor.

When a capacitor AND an inductor are in a circuit, the capacitor supplies leading reactive power to meet the needs of lagging reactive power of the inductor. Essentially, the capacitor acts as a power source for the inductor, decreasing the load on the source. This is the basis of resonance or power factor correction.

Individually, the source would have to supply reactive power to inductors OR capacitors. And reactive power would go back and forth between the devices and the source.

But the collapsing electric field of capacitor creates a magnetic field in inductor and vice versa. $$\V_C\$$ and $$\V_L\$$ can be significantly larger than the source voltage $$\V_S\$$, but you are not getting something for nothing, but rather taking advantage of the fundamental relationships between current and the voltage for the circuit components.

Resonance $$\f_R\$$ is defined as inductive reactance $$\X_L\$$ equal to capacitive reactance $$\X_C\$$. Above $$\f_R\$$, $$\X_L > X_C\$$ and the circuit is inductive. Below $$\f_R\$$, $$\X_C > X_L\$$ and the circuit is capacitive. The circuit has real power P and net reactive power the source has to provide. The capacitor still supplies the inductor, but extra reactive power goes between source and device. Extra leading reactive power probably will not go to the source but find a closer lagging load.

At resonance $$\X_C = X_L\$$ and all of the power required for the inductor comes from the capacitor and all the power provided by the source goes to the resistor. Power factor = 1.