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.

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

Thank you for reading.

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