# Q Factor of RLC Series Resonant Circuit

Regarding the Q Factor of RLC Series Resonant Circuit, this article will explain the information below.

• What is the Q factor of RLC series resonant circuit?
• Derivation of Q factor of RLC series resonant circuit
• Relationship between "Q factor" and "voltage across the inductor L and capacitor C

## What is the Q factor of RLC series resonant circuit?

Q factor is a value that expresses the sharpness of the frequency characteristics. A large Q factor makes the frequency characteristics sharper, while a small Q factor makes the frequency characteristics more gradual.

The above figure shows the frequency characteristics of the magnitude $$I$$ of the current flowing in an RLC series resonant circuit. The RLC series resonant circuit is a circuit consisting of a resistor $$R$$, an inductor $$L$$, and a capacitor $$C$$ connected in series, and the magnitude $$I$$ of current reaches its maximum value $$I_{MAX}$$ at the angular frequency $${\omega}_0\left(=\displaystyle\frac{1}{\sqrt{LC}}\right)$$ where the inductor $$L$$ and capacitor $$C$$ resonate.

In an RLC series resonant circuit, the Q factor is expressed by the following equation (the derivation of the following equation will be explained later).

Q factor of RLC series resonant circuit

\begin{eqnarray}
Q=\frac{{\omega}_0}{{\Delta}{\omega}}=\frac{{\omega}_0}{{\omega}_2-{\omega}_1}=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{{\omega}_0L}{R}=\frac{1}{{\omega}_0CR}\tag{1}
\end{eqnarray}

In the above equation, $${\omega}_0$$ is the resonant angular frequency, $${\omega}_1$$ and $${\omega}_2$$ are the angular frequencies when the magnitude $$I$$ of the current flowing in the RLC series resonant circuit is $$\displaystyle\frac{1}{\sqrt{2}}$$ times the magnitude of $$I_{MAX}$$ ($${\omega}_1{<}{\omega}_2$$), $$R$$ is the resistance of the resistor, $$L$$ is the inductance of the inductor, and $$C$$ is the capacitance of the capacitor.

From equation (1), the smaller $${\Delta}{\omega}(={\omega}_2-{\omega}_1)$$ (the sharper the frequency response), the larger the Q factor.

Supplement

The Q factor is a measure of the low loss of a resonant circuit. In the case of an RLC series resonant circuit, the smaller the resistance $$R$$, the larger the Q factor. This is because the smaller the resistance $$R$$, the smaller the loss generated by the resistor.

## Derivation of Q factor of RLC series resonant circuit

The RLC resonant series circuit is an electrical circuit consisting of a resistor $$R$$, an inductor $$L$$, and a capacitor $$C$$ connected in series.

When the resistance of resistor $$R$$ is $$R{\mathrm{[{\Omega}]}}$$, the inductance of inductor $$L$$ is $$L{\mathrm{[H]}}$$, and the capacitance of capacitor $$C$$ is $$C{\mathrm{[F]}}$$, the impedance $${\dot{Z}}$$ of the RLC series resonant circuit becomes the following equation.

\begin{eqnarray}
{\dot{Z}}=R+j\left({\omega}L-\frac{1}{{\omega}C}\right)\tag{2}
\end{eqnarray}

, where $${\omega}$$ is the angular frequency, which is equal to $$2{\pi}f$$.

The magnitude $$Z$$ of the impedance of the RLC series resonant circuit is expressed by the following equation.

\begin{eqnarray}
Z=|{\dot{Z}}|=\sqrt{R^2+\left({\omega}L-\displaystyle\frac{1}{{\omega}C}\right)^2}\tag{3}
\end{eqnarray}

Therefore, when the magnitude of the supply voltage is $$V$$, the magnitude $$I$$ of the current flowing in the RLC series resonant circuit is expressed by the following equation.

\begin{eqnarray}
I=\frac{V}{Z}=\frac{V}{\sqrt{R^2+\left({\omega}L-\displaystyle\frac{1}{{\omega}C}\right)^2}}\tag{4}
\end{eqnarray}

At the angular frequency (resonant angular frequency) )$${\omega}_0\left(=\displaystyle\frac{1}{\sqrt{LC}}\right)$$, the inductor $$L$$ and the capacitor $$C$$ are in resonance, and the following equation holds.

\begin{eqnarray}
{\omega}_0&=&\frac{1}{\sqrt{LC}}\\
\\
{\Leftrightarrow}{{\omega}_0}^2&=&\frac{1}{LC}\\
\\
{\Leftrightarrow}{{\omega}_0}L&=&\frac{1}{{{\omega}_0}C}\\
\\
{\Leftrightarrow}{{\omega}_0}L-\frac{1}{{{\omega}_0}C}&=&0\tag{5}
\end{eqnarray}

The reactance $$X_L$$ of the inductor $$L$$ is $$X_L={\omega}L$$. On the other hand, the reactance $$X_C$$ of the capacitor $$C$$ is $$X_C=\displaystyle\frac{1}{{\omega}C}$$. As can be seen from equation (5), "reactance $$X_L$$ of inductor $$L$$" and "reactance $$X_C$$ of capacitor $$C$$" are equal at the resonant angular frequency $${\omega}_0$$.

At resonant angular frequency $${\omega}_0$$, the magnitude $$I$$ of the current flowing in the RLC series circuit is the maximum value $$I_{MAX}$$. $$I_{MAX}$$ is expressed by the following equation.

\begin{eqnarray}
I_{MAX}&=&\frac{V}{\sqrt{R^2+\left({\omega}_0L-\displaystyle\frac{1}{{\omega}_0C}\right)^2}}\\
\\
&=&\frac{V}{\sqrt{R^2+0^2}}\\
\\
&=&\frac{V}{R}\tag{6}
\end{eqnarray}

From equations (4) and (6), we obtain the following equation.

\begin{eqnarray}
\frac{I}{I_{MAX}}&=&\frac{\displaystyle\frac{V}{\sqrt{R^2+\left({\omega}L-\displaystyle\frac{1}{{\omega}C}\right)^2}}}{\displaystyle\frac{V}{R}}\\
\\
&=&\frac{R}{\sqrt{R^2+\left({\omega}L-\displaystyle\frac{1}{{\omega}C}\right)^2}}\\
\\
&=&\frac{1}{\sqrt{1+\left(\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}\right)^2}}\tag{7}
\end{eqnarray}

Here, when "the magnitude $$I$$ of the current flowing in the RLC series resonance circuit" is $$\displaystyle\frac{1}{\sqrt{2}}$$ times "the magnitude $$I_{MAX}$$ of the current flowing at resonance" (when $$I=\displaystyle\frac{1}{\sqrt{2}}I_{MAX}$$), the following equation is obtained.

\begin{eqnarray}
\frac{\displaystyle\frac{1}{\sqrt{2}}I_{MAX}}{I_{MAX}}&=&\frac{1}{\sqrt{1+\left(\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}\right)^2}}\\
\\
{\Leftrightarrow}\frac{1}{\sqrt{2}}&=&\frac{1}{\sqrt{1+\left(\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}\right)^2}}\\
\\
{\Leftrightarrow}\frac{{\omega}L}{R}-\frac{1}{{\omega}CR}&=&±1\tag{8}
\end{eqnarray}

By using equation (8), the angular frequency $${\omega}$$ can be obtained for the "Case $$\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=1$$" and the "Case $$\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=-1$$".

Case $$\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=1$$

Transforming "$$\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=1$$", we obtain the following equation.

\begin{eqnarray}
\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=1\\
\\
{\Leftrightarrow}LC{\omega}^2-RC{\omega}-1=0\tag{9}
\end{eqnarray}

Solving equation (9), the angular frequency $${\omega}$$ is expressed by the following equation.

\begin{eqnarray}
{\omega}=\frac{RC+\sqrt{(RC)^2+4LC}}{2LC}\tag{10}
\end{eqnarray}

In the above equation, the root (√) is preceded by "+ (plus)" because $${\omega}{>}0$$ is considered. Let $${\omega}$$ in equation (10) be $${\omega}_2$$.

Case $$\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=-1$$

Transforming "$$\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=-1$$", we obtain the following equation.

\begin{eqnarray}
\displaystyle\frac{{\omega}L}{R}-\displaystyle\frac{1}{{\omega}CR}=-1\\
\\
{\Leftrightarrow}LC{\omega}^2+RC{\omega}-1=0\tag{11}
\end{eqnarray}

Solving equation (11), the angular frequency $${\omega}$$ is expressed by the following equation.

\begin{eqnarray}
{\omega}=\frac{-RC+\sqrt{(RC)^2+4LC}}{2LC}\tag{12}
\end{eqnarray}

In the above equation, the root (√) is preceded by "+ (plus)" because $${\omega}{>}0$$ is considered. Let $${\omega}$$ in equation (12) be $${\omega}_1$$.

Therefore, $${\Delta}{\omega}(={\omega}_2-{\omega}_1)$$ is expressed by the following equation.

\begin{eqnarray}
{\Delta}{\omega}&=&{\omega}_2-{\omega}_1\\
\\
&=&\frac{RC+\sqrt{(RC)^2+4LC}}{2LC}-\frac{-RC+\sqrt{(RC)^2+4LC}}{2LC}\\
\\
&=&\frac{R}{L}\tag{13}
\end{eqnarray}

Note that $${\Delta}{\omega}$$ is called Bandwidth. From the above, the Q factor of the RLC series resonant circuit is expressed by the following equation.

\begin{eqnarray}
Q=\frac{{\omega}_0}{{\Delta}{\omega}}=\frac{{\omega}_0}{{\omega}_2-{\omega}_1}&=&\frac{\displaystyle\frac{1}{\sqrt{LC}}}{\displaystyle\frac{R}{L}}\\
\\
&=&\frac{1}{R}\sqrt{\frac{L}{C}}\tag{14}
\end{eqnarray}

In addition, using "$${\omega}_0=\displaystyle\frac{1}{\sqrt{LC}}$$" in equation (14), the following equation is obtained.

\begin{eqnarray}
Q=\frac{1}{R}\sqrt{\frac{L}{C}}=\frac{{\omega}_0L}{R}=\frac{1}{{\omega}_0CR}\tag{15}
\end{eqnarray}

This concludes the derivation of the Q factor of the RLC series resonant circuit; let's look at the Q factor in a little more detail.

Since the current flowing in the RLC series resonant circuit at the resonant angular frequency $${\omega}_0$$ is $$I_{MAX}$$, the "magnitude $$V_R$$ of the voltage across the resistor $$R$$", the "magnitude $$V_L$$ of the voltage across the inductor $$L$$", and the "magnitude $$V_C$$ of the voltage across the capacitor $$C$$" are as follows.

\begin{eqnarray}
V_R&=&R×I_{MAX}\tag{16}\\
\\
V_L&=&{\omega}_0L×I_{MAX}\tag{17}\\
\\
V_C&=&\frac{1}{{\omega}_0C}×I_{MAX}\tag{18}
\end{eqnarray}

From equations (16) to (18), the Q factor can be transformed as follows.

\begin{eqnarray}
Q&=&\frac{{\omega}_0L}{R}=\frac{{\omega}_0L×I_{MAX}}{R×I_{MAX}}=\frac{V_L}{V_R}\tag{19}\\
\\
Q&=&\frac{1}{{\omega}_0CR}=\frac{{\omega}_0L×I_{MAX}}{R×I_{MAX}}=\frac{V_C}{V_R}\tag{20}
\end{eqnarray}

Therefore, the Q factor can be said to be the ratio of "the magnitude $$V_R$$ of the voltage across the resistor $$R$$" to "the magnitude $$V_L$$ of the voltage across the inductor $$L$$" or "the magnitude $$V_C$$ of the voltage across the capacitor $$C$$"

### Relationship between "Q factor" and "voltage across the inductor L and capacitor C

Since the magnitude of the current flowing in the RLC series resonant circuit at the resonant angular frequency $${\omega}_0$$ is $$I_{MAX}$$, the magnitude $$V_L$$ of the voltage across the inductor $$L$$ is given by the following equation.

\begin{eqnarray}
V_L={\omega}_0L×I_{MAX}={\omega}_0L×\frac{V}{R}=\frac{{\omega}_0L}{R}×V=Q×V\tag{21}
\end{eqnarray}

Similarly, the magnitude $$V_C$$ of the voltage across the capacitor $$C$$ is given by the following equation.

\begin{eqnarray}
V_C=\frac{1}{{\omega}_0C}×I_{MAX}=\frac{1}{{\omega}_0C}×\frac{V}{R}=\frac{1}{{\omega}_0CR}×V=Q×V\tag{22}
\end{eqnarray}

Therefore, the Q factor is also a value that represents how many times the magnitude $$V$$ of the supply voltage is applied to the inductor $$L$$ or the capacitor $$C$$. The larger the Q factor, the larger the voltage applied to the inductor $$L$$ and the capacitor $$C$$.