• Tags: resonance, frequency, Q-factor
• Difficulty: Advanced

Resonance occurs when $X_L$ = $X_C$, i.e., $omega_0$L = $\frac{1}{omega_0*C}$. Solving: $omega_0$ = 1/sqrt(LC), $f_0$ = $\frac{1}{2*pi*sqrt(LC}$). At resonance: (1) Z = R (minimum possible impedance), (2) I = $\frac{V}{R}$ (maximum current), (3) phi = 0 (purely resistive behavior), (4) $V_L$ = $V_C$ = QV (can exceed source voltage). The quality factor Q = $omega_0$*L/R = $\frac{1}{omega_0*C*R}$ = $\frac{1}{R}$*sqrt$\frac{L}{C}$ measures sharpness of resonance. High Q means a narrow, tall resonance peak — the circuit is highly selective (used in radio tuning). Bandwidth $Delta_{omega}$ = $\frac{omega_0}{Q}$ = $\frac{R}{L}$ is the frequency range where current exceeds $I_{max}$/sqrt(2). Increasing R broadens the peak and reduces $I_{max}$. The resonant frequency is independent of R. JEE often asks: how does changing L, C, or R affect the resonance curve?