Resonance in RLC Circuits

Resonance is the tendency of a system to oscillate with greater amplitude at some frequencies than at others. Frequencies at which the response amplitude is a relative maximum are known as the system's resonance frequencies. To study the resonance in an RLC circuit, as illustrated below, we can see how the circuit behaves as a function of the frequency of the driving voltage source.

RLC Series Circuit

An RLC series circuit with an AC voltage source. f is the frequency of the source.

Combining Ohm's law, I_rms=V_rms/Z, and the expression for impedance Z from

$Z = \sqrt{R^2 + (X_L - X_C)^2}$ gives

$I_{rms} = \frac{V_{rms}}{\sqrt{R^2 + (X_L - X_C)^2}}$,

where I_rms and V_rms are rms current and voltage, respectively. The reactances vary with frequency $\nu$, with X_L large at high frequencies and X_C large at low frequencies given as:

$X_L = 2\pi \nu L, X_C = \frac{1} {2\pi \nu C}$.

At some intermediate frequency $\nu_0$, the reactances will be equal and cancel, giving Z=R —this is a minimum value for impedance, and a maximum value for I_rms results. We can get an expression for $\nu_0$ by taking X_L=X_C. Substituting the definitions of X_L and X_C yields:

$\nu_0 = \frac{1}{2\pi \sqrt{LC}}$.

$\nu_0$ is the resonant frequency of an RLC series circuit. This is also the natural frequency at which the circuit would oscillate if not driven by the voltage source. At $\nu_0$, the effects of the inductor and capacitor cancel, so that Z=R, and I_rms is a maximum. Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined as a forced oscillation (in this case, forced by the voltage source) at the natural frequency of the system.

The receiver in a radio is an RLC circuit that oscillates best at its $\nu_0$. A variable capacitor is often used to adjust the resonance frequency to receive a desired frequency and to reject others. is a graph of current as a function of frequency, illustrating a resonant peak in I_rms at $\nu_0 = f_0$. The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus higher-resistance circuits do not resonate as strongly, nor would they be as selective in, for example, a radio receiver.

Current vs. Frequency

A graph of current versus frequency for two RLC series circuits differing only in the amount of resistance. Both have a resonance at f0, but that for the higher resistance is lower and broader. The driving AC voltage source has a fixed amplitude V0.