RLC Series Circuit (Impedance, Phasor Diagram)

Regarding the RLC series circuit, this article will explain the information below.

• Equation, magnitude, vector diagram, and impedance phase angle of RLC series circuit impedance

Impedance of the RLC series circuit

An RLC series circuit (also known as an RLC filter or RLC network) is an electrical circuit consisting of a resistor \(R\), an inductor \(L\), and a capacitor \(C\) connected in series, driven by a voltage source or current source.

The impedance \({\dot{Z}}_R\) of the resistor \(R\), the impedance \({\dot{Z}}_L\) of the inductor \(L\), and the impedance \({\dot{Z}}_C\) of the capacitor \(C\) can be expressed by the following equations:

\begin{eqnarray}
{\dot{Z}}_R&=&R\\
\\
{\dot{Z}}_L&=&jX_L=j{\omega}L\\
\\
{\dot{Z}}_C&=&-jX_C=-j\frac{1}{{\omega}C}=\frac{1}{j{\omega}C}
\end{eqnarray}

,where \({\omega}\) is the angular frequency, which is equal to \(2{\pi}f\), and \(X_L\left(={\omega}L\right)\) is called inductive reactance, which is the resistive component of inductor \(L\) and \(X_C\left(=\displaystyle\frac{1}{{\omega}C}\right)\) is called capacitive reactance, which is the resistive component of capacitor \(C\).

The impedance \({\dot{Z}}\) of the RLC series circuit is the sum of the respective impedance, and is as follow:

\begin{eqnarray}
{\dot{Z}}&=&{\dot{Z}}_R+{\dot{Z}}_L+{\dot{Z}}_C\\
\\
&=&R+jX_L-jX_C\\
\\
&=&R+j\left(X_L-X_C\right)\\
\\
&=&R+j\left({\omega}L-\frac{1}{{\omega}C}\right)
\end{eqnarray}

The impedance \({\dot{Z}}\) can be divided into the following three cases, depending on the size of \(X_L\) and \(X_C\).

• In Case \(X_L{\;}{\gt}{\;}X_C\)
  • The impedance \({\dot{Z}}\) is positive(\({\dot{Z}}{>}0\)) and inductive.
• In Case \(X_L{\;}{\lt}{\;}X_C\)
  • The impedance \({\dot{Z}}\) is negative(\({\dot{Z}}{<}0\)) and capacitive.
• In Case \(X_L=X_C\)
  • The impedance \({\dot{Z}}\) is "\({\dot{Z}}=R\)". In this case, the circuit is in series resonance. When series resonance is established, the angular frequency \({\omega}\) and frequency \(f\) are as follows:
    \begin{eqnarray}
    X_L&=&X_C\\
    \\
    {\omega}L&=&\frac{1}{{\omega}C}\\
    \\
    {\Leftrightarrow}{\omega}&=&\frac{1}{\sqrt{LC}}\\
    \\
    {\Leftrightarrow}f&=&\frac{1}{2{\pi}\sqrt{LC}}\\
    \end{eqnarray}

Magnitude of the impedance of the RLC series circuit

The magnitude \(Z\) of the impedance \({\dot{Z}}\) of the RLC series circuit is the absolute value of  "\({\dot{Z}}=R+j\left({\omega}L-\displaystyle\frac{1}{{\omega}C}\right)\)".

In more detail, the magnitude \(Z\) of the impedance \({\dot{Z}}\) can be obtained by adding the square of the real part \(R\) and the square of the imaginary part \({\omega}L-\displaystyle\frac{1}{{\omega}C}\) and taking the square root, which can be expressed in the following equation.

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

The magnitude \(Z_R\) of the impedance of the resistor \(R\), the magnitude \(Z_L\) of the impedance of the inductor \(L\), and the magnitude \(Z_C\) of the impedance of the capacitor \(C\) are expressed as follows.

\begin{eqnarray}
Z_R&=&|{\dot{Z}}_R|=\sqrt{R^2}=R\\
\\
Z_L&=&|{\dot{Z}}_L|=\sqrt{({\omega}L)^2}={\omega}L\\
\\
Z_C&=&|{\dot{Z}}_C|=\sqrt{\left(\displaystyle\frac{1}{{\omega}C}\right)^2}=\displaystyle\frac{1}{{\omega}C}
\end{eqnarray}

Vector diagram of the RLC series circuit

The vector diagram of the impedance \({\dot{Z}}\) of the RLC series circuit can be drawn in the following steps.

Let's take a look at each step in turn.

Draw a vector of impedance \({\dot{Z}}_R\) of resistor \(R\)

The impedance \({\dot{Z}}_R\) of the resistor \(R\) is expressed as "\({\dot{Z}}_R=R\)".

Therefore, the vector direction of the impedance \({\dot{Z}}_R\) is the direction of the real axis. How to determine the vector orientation will be explained in more detail later.

The magnitude (length) \(Z_R\) of the vector of the impedance \({\dot{Z}}_R\) is "\(Z_R=|{\dot{Z}}_R|=R\)".

Draw a vector of impedance \({\dot{Z}}_L\) of inductor \(L\)

The impedance \({\dot{Z}}_L\) of the inductor \(L\) is expressed as "\({\dot{Z}}_L=j{\omega}L\)".

Therefore, the impedance \({\dot{Z}}_L\) vector is 90° counterclockwise around the real axis (with "\(+j\)", it rotates 90° counterclockwise). How to determine the vector orientation will be explained in detail later.

The magnitude (length) \(Z_L\) of the vector of the impedance \({\dot{Z}}_L\) is "\(Z_L=|{\dot{Z}}_L|={\omega}L\)".

Draw a vector of impedance \({\dot{Z}}_C\) of capacitor \(C\)

The impedance \({\dot{Z}}_C\) of the capacitor \(C\) is expressed as "\({\dot{Z}}_C=-j\displaystyle\frac{1}{{\omega}C}\)".

Therefore, the orientation of the impedance \({\dot{Z}}_C\) vector is 90° clockwise around the real axis (with "\(-j\)", it rotates 90° clockwise). How to determine the vector orientation will be explained in detail later.

The magnitude (length) \(Z_C\) of the vector of the impedance \({\dot{Z}}_C\) is "\(Z_C=|{\dot{Z}}_C|=\displaystyle\frac{1}{{\omega}C}\)".

Combine the vectors

Combining the vector of "impedance \({\dot{Z}}_R\) of resistor \(R\)", "impedance \({\dot{Z}}_L\) of inductor \(L\)", and "impedance \({\dot{Z}}_C\) of capacitor \(C\)" is the vector diagram of the impedance \({\dot{Z}}\) of the RLC series circuit.

The impedance \({\dot{Z}}\) of the RLC series circuit is the sum of the respective impedance, and is as follow:

\begin{eqnarray}
{\dot{Z}}&=&{\dot{Z}}_R+{\dot{Z}}_L+{\dot{Z}}_C\\
\\
&=&R+jX_L-jX_C\\
\\
&=&R+j\left(X_L-X_C\right)\\
\\
&=&R+j\left({\omega}L-\frac{1}{{\omega}C}\right)
\end{eqnarray}

The magnitude of \(X_L\) and \(X_C\) in the parentheses in the above equation changes the vector direction of the impedance \({\dot{Z}}\).

• In Case \(X_L{\;}{\gt}{\;}X_C\)
  • The vector direction of the impedance \({\dot{Z}}\) is upward to the right.
• In Case \(X_L{\;}{\lt}{\;}X_C\)
  • The vector direction of the impedance \({\dot{Z}}\) is downward to the right.
• In Case \(X_L=X_C\)
  • Since the impedance \({\dot{Z}}\) is "\({\dot{Z}}=R\)", the vector direction is to the right.

The magnitude (length) \(Z\) of the vector of the impedance \({\dot{Z}}\) can be expressed as follows.

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

Vector orientation

Here is a more detailed explanation of how vector orientation is determined.

The impedance \({\dot{Z}}_L\) of inductor \(L\) is represented by "\({\dot{Z}}_L=j{\omega}L\)". Therefore, the direction of vector \({\dot{Z}}_L\) is 90° counterclockwise around the real axis.

The impedance \({\dot{Z}}_C\) of capacitor \(C\) is represented by "\({\dot{Z}}_C=-j\displaystyle\frac{1}{{\omega}C}\)". Therefore, the direction of vector \({\dot{Z}}_C\) is 90° clockwise around the real axis.

Impedance phase angle of the RLC series circuit

The impedance phase angle ({\theta}) varies depending on the size of \(X_L\) and \(X_C\).

• In Case \(X_L{\;}{\gt}{\;}X_C\)
  • The impedance phase angle \({\theta}\) is the following value:
    \begin{eqnarray}
    {\theta}&=&{\tan}^{-1}\left(\frac{X_L-X_C}{R}\right){\mathrm{[rad]}}
    \end{eqnarray}
    The impedance angle ({\theta}) of the RLC series circuit is "positive".
• In Case \(X_L{\;}{\lt}{\;}X_C\)
  • The impedance phase angle \({\theta}\) is the following value:
    \begin{eqnarray}
    {\theta}&=&{\tan}^{-1}\left(\frac{X_L-X_C}{R}\right){\mathrm{[rad]}}
    \end{eqnarray}
    The impedance angle ({\theta}) of the RLC series circuit is "negative".
• In Case \(X_L=X_C\)
  • The impedance phase angle \({\theta}\) is the following value:
    \begin{eqnarray}
    {\theta}=0{\mathrm{[rad]}}
    \end{eqnarray}

Summary

In this article, the following information on "RLC series circuit was explained.

1. Equation, magnitude, vector diagram, and impedance phase angle of RLC series circuit impedance

Thank you for reading.