# Q Factor of RLC Parallel Resonant Circuit

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

• What is the Q factor of RLC parallel resonant circuit?
• Derivation of Q factor of RLC parallel resonant circuit
• Relationship between "Q factor" and "current flowing through the inductor L and capacitor C

## What is the Q factor of RLC parallel 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 parallel resonant circuit. The RLC parallel resonant circuit is a circuit consisting of a resistor $$R$$, an inductor $$L$$, and a capacitor $$C$$ connected in parallel, and the magnitude $$I$$ of current reaches its minimum value $$I_{MIN}$$ at the angular frequency $${\omega}_0\left(=\displaystyle\frac{1}{\sqrt{LC}}\right)$$ where the inductor $$L$$ and capacitor $$C$$ resonate.

In an RLC parallel 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 parallel resonant circuit

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

In the above equation, $${\omega}_0$$ is the resonant angular frequency, $$R$$ 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 parallel resonant circuit, the larger the resistance $$R$$, the larger the Q factor. This is because the larger the resistance $$R$$, the smaller the current $$I_R$$ flowing through the resistor and the smaller the loss generated by the current $$I_R$$ flowing through the resistor.

## Derivation of Q factor of RLC parallel resonant circuit

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

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 parallel resonant circuit becomes the following equation.

The definition of Q factor in an RLC parallel resonant circuit is the ratio of "the magnitude $$I_L$$ of the current flowing in the inductor $$L$$ (or the magnitude $$I_C$$ of the current flowing in the capacitor $$C$$) at resonance" and "the magnitude $$I_R$$ of the current flowing in the resistor $$R$$ at resonance".

If this definition is written in an equation, the Q factor of the RLC parallel resonance circuit is expressed by the following equation.

\begin{eqnarray}
Q=\frac{I_L}{I_R}=\frac{I_C}{I_R}\tag{2}
\end{eqnarray}

Now, let's find "the magnitude $$I_R$$ of the current flowing in the resistor $$R$$", "the magnitude $$I_L$$ of the current flowing in the inductor $$L$$", and "the magnitude $$I_C$$ of the current flowing in the capacitor $$C$$" at the resonance angular frequency $${\omega}_0$$.

• The magnitude $$I_R$$ of the current flowing in the resistor $$R$$
• Since the magnitude $$V_R$$ of the voltage across the resistor $$R$$ is equal to the magnitude $$V$$ of the power supply voltage, the magnitude $$I_R$$ of the current flowing through the resistor $$R$$ is expressed by the following equation.
\begin{eqnarray}
I_R=\frac{V_R}{R}=\frac{V}{R}\tag{3}
\end{eqnarray}
• The magnitude $$I_L$$ of the current flowing in the inductor $$L$$
• The magnitude $$V_L$$ of the voltage across the inductor $$L$$ is equal to the magnitude $$V$$ of the power supply voltage. The magnitude $$X_L$$ of the reactance of inductor $$L$$ is expressed by "$$X_L={\omega}L$$". Therefore, the magnitude $$I_L$$ of the current flowing through the inductor $$L$$ at the resonant angular frequency $${\omega}_0$$ is expressed by the following equation.
\begin{eqnarray}
I_L=\frac{V_L}{X_L}=\frac{V}{{\omega}_0L}\tag{4}
\end{eqnarray}
• The magnitude $$I_C$$ of the current flowing in the capacitor $$C$$
• The magnitude $$V_C$$ of the voltage across the capacitor $$C$$ is equal to the magnitude $$V$$ of the power supply voltage. The magnitude $$X_C$$ of the reactance of capacitor $$C$$ is expressed by "$$X_C=\displaystyle\frac{1}{{\omega}C}$$". Therefore, the magnitude $$I_C$$ of the current flowing through the capacitor $$C$$ at the resonant angular frequency $${\omega}_0$$ is expressed by the following equation.
\begin{eqnarray}
I_C=\frac{V_C}{X_C}=\frac{V}{\displaystyle\frac{1}{{\omega}_0C}}={\omega}_0CV\tag{5}
\end{eqnarray}

From equations (2)-(5), the Q factor of the RLC parallel resonant circuit can be expressed by the following equation.

\begin{eqnarray}
Q&=&\frac{I_L}{I_R}=\frac{\displaystyle\frac{V}{{\omega}_0L}}{\displaystyle\frac{V}{R}}=\frac{R}{{\omega}_0L}\tag{6}\\
\\
Q&=&\frac{I_C}{I_R}=\frac{{\omega}_0CV}{\displaystyle\frac{V}{R}}={\omega}_0CR\tag{7}\\
\end{eqnarray}

In addition, using "$${\omega}_0=\displaystyle\frac{1}{\sqrt{LC}}$$" in equations (6) or (7), the following equation is obtained.

\begin{eqnarray}
Q=R\sqrt{\frac{C}{L}}\tag{8}
\end{eqnarray}

This concludes the derivation of the Q factor of the RLC parallel resonant circuit.

### Relationship between "Q factor" and "current flowing through the inductor L and capacitor C

From equation (6), the magnitude $$I_L$$ of the current flowing in the inductor $$L$$ at the resonant angular frequency $${\omega}_0$$ can be expressed by the following equation

\begin{eqnarray}
I_L=Q×I_R\tag{9}
\end{eqnarray}

Similarly, the magnitude $$I_C$$ of the current flowing through the capacitor $$C$$ is given by the following equation.

\begin{eqnarray}
I_C=Q×I_R\tag{10}
\end{eqnarray}

Therefore, the Q factor is also a value that expresses how many times the magnitude $$I_R$$ of the current flowing through the resistor $$R$$ flows through the inductor $$L$$ or the capacitor $$C$$. The larger the Q factor, the larger the current flowing through the inductor $$L$$ and the capacitor $$C$$.

I will now explain the RLC parallel resonance circuit in detail. At the beginning of this section, we explained that "at the angular frequency $${\omega}_0\left(=\displaystyle\frac{1}{\sqrt{LC}}\right)$$ where the inductor $$L$$ and the capacitor $$C$$ resonate, the magnitude of the current $$I$$ flowing in the RLC parallel resonance circuit reaches the minimum value $$I_{MIN}$$. Let us derive this value of $$I_{MIN}$$.

The admittance $${\dot{Y}}$$ of an RLC parallel resonant circuit is expressed by the following equation.

\begin{eqnarray}
{\dot{Y}}&=&\frac{1}{R}+\frac{1}{j{\omega}L}+j{\omega}C\\
\\
&=&\frac{1}{R}+j\left({\omega}C-\frac{1}{{\omega}L}\right)\tag{11}
\end{eqnarray}

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

The magnitude $$Y$$ of the admittance of the RLC parallel resonant circuit is expressed by the following equation.

\begin{eqnarray}
Y=|{\dot{Y}}|=\sqrt{\left(\frac{1}{R}\right)^2+\left({\omega}C-\displaystyle\frac{1}{{\omega}L}\right)^2}\tag{12}
\end{eqnarray}

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

\begin{eqnarray}
I=VY=V\sqrt{\left(\frac{1}{R}\right)^2+\left({\omega}C-\displaystyle\frac{1}{{\omega}L}\right)^2}\tag{13}
\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}_0C-\displaystyle\frac{1}{{\omega}_0L}&=&0\tag{14}
\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 (14), "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 parallel circuit is the minimum value $$I_{MIN}$$. $$I_{MIN}$$ is expressed by the following equation.

\begin{eqnarray}
I_{MIN}&=&V\sqrt{\left(\frac{1}{R}\right)^2+\left({\omega}_0C-\displaystyle\frac{1}{{\omega}_0L}\right)^2}\\
\\
&=&V\sqrt{\left(\frac{1}{R}\right)^2+0^2}\\
\\
&=&\frac{V}{R}\tag{15}
\end{eqnarray}

If we look at equations (3) and (15), we see that $$I_R$$ and $$I_{MIN}$$ have the same value. In other words, equations (9) and (10) can be transformed into the following equation

\begin{eqnarray}
I_L&=&Q×I_R=Q×I_{MIN}\tag{16}\\
\\
I_C&=&Q×I_R=Q×I_{MIN}\tag{17}
\end{eqnarray}

Therefore, the Q factor is also a value that expresses how many times the magnitude $$I_{MIN}$$ of the current flowing from the power supply flows through the inductor $$L$$ or capacitor $$L$$ at resonance (at the resonance angular frequency $${\omega}_0$$). The larger the Q factor, the larger the current flowing through the inductor $$L$$ and capacitor $$C$$.