Resonance in Series RLC Circuit

Electrical resonance in a series RLC circuit occurs when the frequency $\omega_r$ of the applied alternating source exactly equals the circuit's own NATURAL frequency $1/\sqrt{LC}$, at which point the current in the circuit reaches its overall maximum possible value. The frequency at which this happens is called the resonant frequency,

Since $\omega_r=1/\sqrt{LC}$ means $\omega_r^2=1/LC$, i.e. $\omega_r L=1/(\omega_r C)$, this is exactly the condition

the SAME special-case condition already identified in section 4.7.8: resonance is achievable by varying the applied frequency until the (frequency-dependent) inductive and capacitive reactances become exactly equal. Resonance requires the circuit to contain BOTH L and C together -- only then can $V_L$ and $V_C$, being $180^{\circ}$ out of phase with each other, cancel one another exactly and leave the circuit purely resistive; a circuit with only R and L, or only R and C, therefore can never resonate at any frequency, however it is tuned.

Effects of resonance. At resonance, since $X_L=X_C$, the impedance collapses to its minimum possible value, $Z=\sqrt{R^2+(X_L-X_C)^2}=\sqrt{R^2}=R \qquad(\text{since }X_L=X_C)$, and correspondingly the current reaches its maximum value,

limited purely by the circuit's own resistance R -- a smaller R gives a LARGER peak current and a sharper, more narrowly-peaked resonance curve (current plotted against frequency); a larger R gives a smaller peak current and a broader, less sharply-tuned curve. …