Resonance in a Series LCR Circuit

The resonance condition. Section 7.10.1 showed that the impedance of a series LCR circuit is $Z=\sqrt{R^2+(X_L-X_C)^2}$. Since $R$ is fixed, $Z$ is smallest exactly when the term $(X_L-X_C)^2$ is smallest -- which happens when $X_L=X_C$, making that whole term vanish. At this special condition, called resonance, the impedance collapses to its absolute minimum possible value:

The resonant frequency. Setting $X_L=X_C$ explicitly,

with the corresponding ordinary (cyclic) resonant frequency $f_0=\omega_0/(2\pi)=1/(2\pi\sqrt{LC})$. This frequency depends ONLY on $L$ and $C$ -- not on $R$ at all -- so changing $R$ never shifts WHERE the peak occurs, only how tall and narrow it is (see below).

Why the current is maximum at resonance. Since $I_{rms}=V_{rms}/Z$ for a fixed applied rms voltage, and $Z$ is at its absolute smallest value $R$ exactly at $\omega=\omega_0$, the rms current reaches its own absolute MAXIMUM value at resonance:

Away from $\omega_0$ in either direction (whether $\omega<\omega_0$, where the circuit is net capacitive, or $\omega>\omega_0$, where it is net inductive), $(X_L-X_C)^2$ grows, $Z$ increases above $R$, and the current falls below this peak -- producing the bell-shaped resonance curve shown in the figure for this section.

Why smaller $R$ gives a sharper peak. The PEAK height $I_{max}=V_{rms}/R$ is obviously larger for a smaller $R$. But $R$ also controls how QUICKLY the current falls away as $\omega$ moves away from $\omega_0$: since $Z=\sqrt{R^2+(X_L-X_C)^2}$, a small $R$ means even a modest mismatch $(X_L-X_C)$ already dominates $R^2$ inside the square root, so $Z$ (and hence $I$) changes rapidly with frequency near $\omega_0$ -- giving a tall, NARROW resonance peak. A large $R$, by contrast, dominates the sum for a wider range of $(X_L-X_C)$, so $Z$ changes only gradually with frequency, giving a shorter, BROADER peak (both curves are shown together in the figure, sharing the same $\omega_0$ but with very different shapes). …