Suppose we vary the emf frequency $\omega$ while keeping everything else constant. There is very little current at very low frequencies because the capacitive reactance $X_{\text{C}}=1\text{/}\omega C(\text{and thus Z})$ is very large. Similarly, there is very little current at very high frequencies because the inductive reactance $X_{\text{L}}=\omega L$ becomes very large.

If $I$ approaches zero at very low and very high frequencies, there should be some intermediate frequency where $I$ is a maximum. Indeed, you can see from Equation 32.24 that the denominator will be a minimum, making $I$ a maximum, when $X_{\text{}\text{L}}=X_{\text{}\text{C}}$, or

The frequency ${\omega }_0$ that satisfies Equation 32.29 is called the resonance frequency:

This is the frequency for maximum current in the series $RLC$ circuit. The maximum current

is that of a purely resistive circuit because the impedance is $Z=R$ at resonance.

You’ll recognize ${\omega }_0$ as the oscillation frequency of the $LC$ circuit we analyzed in Chapter 30. The current in an ideal $LC$ circuit oscillates forever as energy is transferred back and forth between the capacitor and the inductor. This is analogous to an ideal, frictionless simple harmonic oscillator in which the energy is transformed back and forth between kinetic and potential.

Adding a resistor to the circuit is like adding damping to a mechanical oscillator. The emf is then a sinusoidal driving force, and the series $RLC$ circuit is directly analogous to the driven, damped oscillator that you studied in Chapter 15. A mechanical oscillator exhibits resonance by having a large-amplitude response when the driving frequency matches the system’s natural frequency. Equation 32.30 is the natural frequency of the series $RLC$ circuit, the frequency at which the current would like to oscillate. Consequently, the circuit has a large current response when the oscillating emf matches this frequency.

FIGURE 32.18 shows the peak current $I$ of a series $RLC$ circuit as the emf frequency $\omega$ is varied. Notice how the current increases until reaching a maximum at frequency ${\omega }_0$, then decreases. This is the hallmark of a resonance.

FIGURE 32.18 A graph of the current $I$ versus emf frequency for a series $RLC$ circuit.

As $R$ decreases, causing the damping to decrease, the maximum current becomes larger and the curve in Figure 32.18 becomes narrower. You saw exactly the same behavior for a driven mechanical oscillator. The emf frequency must be very close to ${\omega }_0$ in order for a lightly damped system to respond, but the response at resonance is very large.

For a different perspective, FIGURE 32.19 graphs the instantaneous emf $\epsilon ={\epsilon }_{\text{}0}\text{ cos }\omega t$ and current $i=I\text{cos}(\omega t-\varphi )$ for frequencies below, at, and above ${\omega }_0$. The current and the emf are in phase at resonance $(\varphi =0\text{rad})$ because the capacitor and inductor essentially cancel each other to give a purely resistive circuit. Away from resonance, the current decreases and begins to get out of phase with the emf. You can see, from Equation 32.27, that the phase angle $\varphi$ is negative when $X_{\text{L}}<X_{\text{C}}$ (i.e., the frequency is below resonance) and positive when $X_{\text{L}}>X_{\text{C}}$ (the frequency is above resonance).

FIGURE 32.19 Graphs of the emf $\epsilon$ and the current $i$ at frequencies below, at, and above the resonance frequency ${\omega }_0$.

Resonance circuits are widely used in radio, television, and communication equipment because of their ability to respond to one particular frequency (or very narrow range of frequencies) while suppressing others. The selectivity of a resonance circuit improves as the resistance decreases, but the inherent resistance of the wires and the inductor coil keeps $R$ from being $0\text{Ω}$.