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}

- 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:

### 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}

Supplement

Some impedance \(Z\) symbols have a ". (dot)" above them and are labeled \({\dot{Z}}\).

\({\dot{Z}}\) with this dot represents a vector.

If it has a dot (e.g. \({\dot{Z}}\)), it represents a vector (complex number), and if it does not have a dot (e.g. \(Z\)), it represents the absolute value (magnitude, length) of the vector.

## 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.

How to draw a Vector Diagram

- Draw a vector of impedance \({\dot{Z}}_R\) of resistor \(R\)
- Draw a vector of impedance \({\dot{Z}}_L\) of inductor \(L\)
- Draw a vector of impedance \({\dot{Z}}_C\) of capacitor \(C\)
- Combine the vectors

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}

Supplement

The magnitude (length) \(Z\) of the vector of the synthetic impedance \({\dot{Z}}\) of the RLC series circuit can also be obtained using the Pythagorean theorem in the vector diagram.

### Vector orientation

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

Vector orientation

When an imaginary unit "\(j\)" is added to the expression, the direction of the vector is rotated by 90°.

- With "\(+j\)" is attached
- The vector rotates 90° counterclockwise.

- With "\(-j\)" is attached
- The vector rotates 90° clockwise.

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".

- The impedance phase angle \({\theta}\) is the following value:
- 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".

- The impedance phase angle \({\theta}\) is the following value:
- In Case \(X_L=X_C\)
- The impedance phase angle \({\theta}\) is the following value:

\begin{eqnarray}

{\theta}=0{\mathrm{[rad]}}

\end{eqnarray}

- The impedance phase angle \({\theta}\) is the following value:

#### Summary

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

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

Thank you for reading.

Related article

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