# RL Series Circuit (Impedance, Phasor Diagram)

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

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

## Impedance of the RL series circuit

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

The impedance $${\dot{Z}}_R$$ of the resistor $$R$$ and the impedance $${\dot{Z}}_L$$ of the inductor $$L$$ can be expressed by the following equations:

\begin{eqnarray}
{\dot{Z}}_R&=&R\\
\\
{\dot{Z}}_L&=&jX_L=j{\omega}L
\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 $$L$$.

The impedance $${\dot{Z}}$$ of the RL series circuit is the sum of the respective impedance, and is as follows:

\begin{eqnarray}
{\dot{Z}}&=&{\dot{Z}}_R+{\dot{Z}}_L\\
\\
&=&R+jX_L\\
\\
&=&R+j{\omega}L
\end{eqnarray}

### Magnitude of the impedance of the RL series circuit

The magnitude $$Z$$ of the impedance $${\dot{Z}}$$ of the RL series circuit is the absolute value of  "$${\dot{Z}}=R+j{\omega}L$$".

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 $${\omega}L$$ and taking the square root, which can be expressed in the following equation:

\begin{eqnarray}
Z=|{\dot{Z}}|=\sqrt{R^2+({\omega}L)^2}
\end{eqnarray}

The magnitude $$Z_R$$ of the impedance of the resistor $$R$$ and the magnitude $$Z_L$$ of the impedance of the inductor $$L$$ 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
\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 RL series circuit

The vector diagram of the impedance $${\dot{Z}}$$ of the RL 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 $$L$$
• 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 orientation of 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 "$$Z_L=|{\dot{Z}}_L|={\omega}L$$".

### Combine the vectors

Combining the vector of "impedance $${\dot{Z}}_R$$ of resistor $$R$$" and "impedance $${\dot{Z}}_L$$ of inductor $$L$$" is the vector diagram of the impedance $${\dot{Z}}$$ of the RL series circuit.

The magnitude (length) $$Z$$ of the vector of the impedance $${\dot{Z}}$$ is "$$Z=|{\dot{Z}}|=\sqrt{R^2+({\omega}L)^2}$$".

Supplement

The magnitude (length) $$Z$$ of the vector of the synthetic impedance $${\dot{Z}}$$ of the RL 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.

## Impedance phase angle of the RL series circuit

The impedance phase angle $${\theta}$$ of the RL series circuit can be obtained from the vector diagram.

The impedance phase angle $${\theta}$$ of the RL series circuit is expressed by the following equation:

\begin{eqnarray}
{\tan}{\theta}&=&\frac{{\omega}L}{R}\\
\\
{\Leftrightarrow}{\theta}&=&{\tan}^{-1}\frac{{\omega}L}{R}\\
\end{eqnarray}

#### Summary

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

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