# 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}

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}
\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}
\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}