• LC series resonant circuit: L and C are in series. The impedance $Z_{TOT}=X_L+X_C=j \omega L - j \frac{1}{\omega C}$, the current $I=\frac{Vs}{Z_{TOT}}$. By converting $\omega$ to the frequency f, we can rewrite the impedance and the current can be represented as a function of f. At the resonant frequency, the impedance goes to zero while the current goes to infinity.
• The physical meaning is that inductive and capacitive reactances at resonance are equal but opposite in phase.
• For an LC parallel-resonant circuit, at the resonant frequency the impedance goes to infinity while the current goes to zero.
• When a resistance is introduced, it prevents a zero impedance condition, The unloaded quality factor ($Q_u$) can be calculated by $\frac{1}{R}\sqrt{\frac{L}{C}} =\frac{X_{L,0}}{R}=\frac{2 \pi f_0 L}{R}$ (remember that at resonance $X_{L,0}$ and $X_{C,0}$ are equal but opposite in phase). The quality factor is independent of any external load, so it is called unloaded.
• The bandwidth can be computed by $BW = \frac{f_0}{Q_u}$.
• For a RLC parallel-resonant circuit, we can compute the parallel-equivalent resistance from a leg with series $L$ and $R_S$ by: $R_P=\frac{X_L^2}{R_S}=\frac{(2 \pi f_0 L)^2}{R_S}=Q_u X_L$.
• The quality factor of loaded circuits can be computed by $Q_{LOAD}=\frac{R_{LOAD}}{X}$.