A series RLC circuit contains a resistor, capacitor and inductor carrying the same sinusoidal current. The three elements respond to that current with different phase relationships, and their competition produces electrical resonance.

Let the current be

$$i(t)=I_m\cos(\omega t-\phi).$$

The resistor voltage is in phase with the current and has amplitude

$$V_R=I_mR.$$

The inductor voltage leads the current by $90^\circ$ and has amplitude

$$V_L=I_mX_L=I_m\omega L.$$

The capacitor voltage lags the current by $90^\circ$ and has amplitude

$$V_C=I_mX_C=\frac{I_m}{\omega C}.$$

The inductor and capacitor contributions are therefore opposite in phase. Their net reactive amplitude is proportional to

$$X_L-X_C=\omega L-\frac1{\omega C}.$$

Current amplitude

Because the resistor contribution is $90^\circ$ out of phase with the net reactive contribution, the source-voltage amplitude satisfies

$$V_m^2=(I_mR)^2+\left[I_m(X_L-X_C)\right]^2.$$

Therefore

$$\boxed{I_m=\frac{V_m}{\sqrt{R^2+(X_L-X_C)^2}}}$$

or

$$\boxed{I_m=\frac{V_m}{\sqrt{R^2+\left(\omega L-\frac1{\omega C}\right)^2}}}.$$

The denominator is the magnitude of the circuit's opposition to the sinusoidal source. In more general AC analysis it is represented compactly by complex impedance, but that representation is not required to understand the resonance itself.

Resonance

For fixed source-voltage amplitude, the current is largest when the reactive contributions cancel:

$$X_L=X_C.$$

Thus

$$\omega_0L=\frac1{\omega_0C},$$

so

$$\boxed{\omega_0=\frac1{\sqrt{LC}}}.$$

At this frequency the source only has to balance the resistor voltage:

$$V_m=I_mR,$$

and the current reaches

$$\boxed{I_{m,\max}=\frac{V_m}{R}}.$$

The source voltage and current are then in phase.

Worked example

Let

$$L=20,\mathrm{mH},\qquad C=5.0,\mu\mathrm F,\qquad R=10,\Omega.$$

Then

$$\omega_0=\frac1{\sqrt{(20\times10^{-3})(5.0\times10^{-6})}} \approx3162,\mathrm{rad/s}.$$

The resonant frequency is

$$f_0=\frac{\omega_0}{2\pi}\approx503,\mathrm{Hz}.$$

If the source amplitude is

$$V_m=5.0,\mathrm V,$$

then at resonance

$$I_{m,\max}=\frac{5.0}{10}=0.50,\mathrm A.$$

Below and above resonance

At low frequency,

$$X_C>X_L,$$

so the circuit is predominantly capacitive and the current leads the source voltage.

At high frequency,

$$X_L>X_C,$$

so the circuit is predominantly inductive and the current lags the source voltage.

The phase crosses through zero at resonance.

Energy exchange and damping

The capacitor stores energy in its electric field while the inductor stores energy in its magnetic field. Near resonance, energy is exchanged repeatedly between these two storage mechanisms.

The resistor removes energy from the oscillation as heat. A smaller resistance generally produces a taller and narrower resonance peak, while larger resistance broadens and suppresses the response.

Series RLC resonance is the electrical counterpart of a driven mechanical oscillator: inductance provides an inertia-like effect, capacitance provides a restoring storage mechanism, and resistance provides damping.