Consider the series RLC circuit shown in Fig. 1. The impedance seen by the voltage source is $Z=R+\textcolor{blue}{j} \left(\omega L + \frac{1}{\omega C}\right)$. The frequency at which the impedance is purely real is called resonance frequency $$\begin{equation}\omega_0=\frac{1}{\sqrt{L C}}\end{equation}$$ At resonance (when $\omega=\omega_0$), the magnitude of current $$\begin{equation}I=\frac{V}{R+\textcolor{blue}{j} \left(\omega L + \frac{1}{\omega C}\right)}\end{equation}$$ is maximum.

The quality factor of the series RLC circuit is defined as the magnitude of the ratio of the voltage across the inductor (or capacitor) to the voltage across the resistor at resonance $$\begin{equation}Q=\frac{\left|V_L\right|}{\left|V_R\right|}=\frac{\left|V_C\right|}{\left|V_R\right|}=\frac{\omega_0 L}{R}=\frac{1}{\omega_0 C R}=\frac{1}{R}{\sqrt{\frac{L}{C}}}\end{equation}$$

Fig. 2 shows the relative magnitude of $\left|\frac{V_R}{V_1}\right|$ as a function of frequency. If the resistor denotes the output and the voltage source is the input, the RLC circuit shown in Fig. 1 is a band-pass filter, in which the maximum of the transfer function occurs at the resonant frequency $\omega_0$ and the bandwidth is $$\begin{equation}BW=\omega_H-\omega_L\end{equation}$$ where $\omega_H$ and $\omega_L$ are the half-power frequencies of the filter. The half-power frequencies are defined as the frequencies at which the relative magnitude of the transfer function decreases to $\frac{1}{\sqrt{2}}$ and can be computed by solving $\left|\frac{V_R}{V_1}\right| = \frac{R}{\sqrt{R^2+ \left(\omega L + \frac{1}{\omega C}\right)^2}}=\frac{1}{\sqrt{2}}$. We obtain $$\begin{equation}\omega_L=\omega_0\left(-\frac{1}{2 Q}+\sqrt{\frac{1}{4 Q^2}+1}\right) \end{equation}$$ $$\begin{equation}\omega_H=\omega_0\left(\frac{1}{2 Q}+\sqrt{\frac{1}{4 Q^2}+1}\right) \end{equation}$$ Multiplying the last two equations, we find $$\begin{equation}\omega_0^2=\omega_L\omega_H\end{equation}$$ which shows that the resonant frequency $\omega_0$ is the geometric mean of the two half-power frequencies.

Although the magnitude of the current in the circuit (and consequently the magnitude of the voltage across the resistor) is maximum when $\omega=\omega_0$, it can be shown that the voltage across the reactive elements is not maximum at resonance. Indeed, the the frequency at which the voltage across the inductor is maximum can be computed by maximizing $\left|\frac{V_L}{V}\right| = \frac{\left|\textcolor{blue}{j} \omega L\right|}{\left|R+\textcolor{blue}{j} \left(\omega L + \frac{1}{\omega C}\right)\right|} = \frac{\omega L}{\sqrt{R^2+ \left(\omega L + \frac{1}{\omega C}\right)^2}}$ with respect to $\omega$, which gives $$\begin{equation}\omega_{max,L}=\omega_0\left(1-\frac{1}{2 Q^2}\right)^{-\frac{1}{2}}\end{equation}$$ The frequency at which the voltage across the capacitor is maximum can be computed by maximizing $\left|\frac{V_C}{V}\right| = \frac{1}{\omega C \left|R+\textcolor{blue}{j} \left(\omega L + \frac{1}{\omega C}\right)\right|} = \frac{1}{\omega C \sqrt{R^2+ \left(\omega L + \frac{1}{\omega C}\right)^2}}$ with respect to $\omega$, which gives $$\begin{equation}\omega_{max,C}=\omega_0\left(1-\frac{1}{2 Q^2}\right)^{\frac{1}{2}}\end{equation}$$ Notice that the above frequencies exist only when $Q \gt \sqrt{2}$. In addition, the following equation is satisfied $$\begin{equation}\omega_0^2=\omega_{max,L}\cdot\omega_{max,C}\end{equation}$$ For large $Q$ we have $\omega_{max,L}\approx\omega_0\approx \omega_{max,C}$.

Now, consider the parallel RLC circuit shown in Fig. 3. The impedance seen by the current source is $Z=\left[\frac{1}{R}+\textcolor{blue}{j} \left(\omega C + \frac{1}{\omega L}\right)\right]^{-1}$ is pureley real when the frequency is equal to the resonance frequency $$\begin{equation}\omega_0=\frac{1}{\sqrt{L C}}\end{equation}$$ At resonance, the magnitude of the voltage in across the components $$\begin{equation}\V=\frac{I}{\frac{1}{R}+\textcolor{blue}{j} \left(\omega C + \frac{1}{\omega L}\right)\}\end{equation}$$ is maximum.

The quality factor of the paralel RLC circuit is defined as the magnitude of the ratio of the current going through the inductor (or capacitor) to the current going through the resistor at resonance $$\begin{equation}Q=\frac{\left|I_L\right|}{\left|I_R\right|}=\frac{\left|I_C\right|}{\left|I_R\right|}=\frac{R}{\omega_0 L}=\omega_0 C R=R \sqrt{\frac{C}{L}}\end{equation}$$ (notice that this result is the reciprocal of $Q$ for the series case).

If the resistor denotes the output and the current source is the input, the RLC circuit can be regarded as a band-stop filter, in which the maximum of the transfer function, $\left|\frac{V_R}{I_1}\right|$, occurs at the resonant frequency $\omega_0$ and the bandwidth is $$\begin{equation}BW=\omega_H-\omega_L\end{equation}$$ where $\omega_H$ and $\omega_L$ are the half-power frequencies of the filter, which are given by $$\begin{equation}\omega_L=\omega_0\left(-\frac{1}{2 Q}+\sqrt{\frac{1}{4 Q^2}+1}\right) \end{equation}$$ $$\begin{equation}\omega_H=\omega_0\left(\frac{1}{2 Q}+\sqrt{\frac{1}{4 Q^2}+1}\right) \end{equation}$$ One can verify that $$\begin{equation}\omega_0^2=\omega_L\omega_H\end{equation}$$

The frequencies at which the currents going through the capacitor and the inductor are maximum are $$\begin{equation}\omega_{max,L}=\omega_0\left(1-\frac{1}{2 Q^2}\right)^{\frac{1}{2}}\end{equation}$$ $$\begin{equation}\omega_{max,C}=\omega_0\left(1-\frac{1}{2 Q^2}\right)^{-\frac{1}{2}}\end{equation}$$ Similar to the series case, these frequencies exist only when $Q \gt \sqrt{2}$ and satisfy $\omega_0^2=\omega_{max,L}\cdot\omega_{max,C}$.