# Resonance in RLC Series AC Circuits

## 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.

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