## Resonance in *RLC* Series AC Circuits

How does an *RLC* circuit behave as a function of the frequency of the driving voltage source? Combining Ohm’s law, \({I}_{\text{rms}}={V}_{\text{rms}}/Z\), and the expression for impedance \(Z\) from \(Z=\sqrt{{R}^{2}+({X}_{L}-{X}_{C}{)}^{2}}\) gives

\({I}_{\text{rms}}=\cfrac{{V}_{\text{rms}}}{\sqrt{{R}^{2}+({X}_{L}-{X}_{C}{)}^{2}}}\text{.}\)

The reactances vary with frequency, with \({X}_{L}\) large at high frequencies and \({X}_{C}\) large at low frequencies, as we have seen in three previous examples. At some intermediate frequency \({f}_{0}\), the reactances will be equal and cancel, giving *\(Z=R\)* —this is a minimum value for impedance, and a maximum value for \({I}_{\text{rms}}\) results. We can get an expression for \({f}_{0}\) by taking

\({X}_{L}={X}_{C}\text{.}\)

Substituting the definitions of \({X}_{L}\) and \({X}_{C}\),

\(2{\mathrm{\pi f}}_{0}L=\cfrac{1}{2{\mathrm{\pi f}}_{0}C}\text{.}\)

Solving this expression for \({f}_{0}\) yields

\({f}_{0}=\cfrac{1}{2\pi \sqrt{\text{LC}}}\text{,}\)

where \({f}_{0}\) 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 \({f}_{0}\), the effects of the inductor and capacitor cancel, so that *\(Z=R\)*, and \({I}_{\text{rms}}\) is a maximum.

Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be 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 \({f}_{0}\). A variable capacitor is often used to adjust \({f}_{0}\) to receive a desired frequency and to reject others. This figure is a graph of current as a function of frequency, illustrating a resonant peak in \({I}_{\text{rms}}\) at \({f}_{0}\). 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 the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.