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

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

## Impedance of the RC series circuit

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

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

\begin{eqnarray}

{\dot{Z}}_R&=&R\\

\\

{\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 \(X_C\left(=\displaystyle\frac{1}{{\omega}C}\right)\) is called capacitive reactance, which is the resistive component of capacitor \(C\).

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

\begin{eqnarray}

{\dot{Z}}&=&{\dot{Z}}_R+{\dot{Z}}_C\\

\\

&=&R+\frac{1}{j{\omega}C}\\

\\

&=&R-j\frac{1}{{\omega}C}

\end{eqnarray}

### Magnitude of the impedance of the RC series circuit

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

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 \(\displaystyle\frac{1}{{\omega}C}\) and taking the square root, which can be expressed in the following equation:

\begin{eqnarray}

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

\end{eqnarray}

The magnitude \(Z_R\) of the impedance of the resistor \(R\) 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_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 RC series circuit

The vector diagram of the impedance \({\dot{Z}}\) of the RC 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 \({\dot{Z}}_C\) 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 \({\dot{Z}}_C\) 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 \(R\)" and "impedance \({\dot{Z}}_C\) of capacitor \(C\)" is the vector diagram of the impedance \({\dot{Z}}\) of the RC series circuit.

The magnitude (length) \(Z\) of the vector of the impedance \({\dot{Z}}\) is "\(Z=|{\dot{Z}}|=\displaystyle\sqrt{R^2+\left(\displaystyle\frac{1}{{\omega}C}\right)^2}\)".

Supplement

The magnitude (length) \(Z\) of the vector of the synthetic impedance \({\dot{Z}}\) of the RC 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 \({\dot{Z}}_C\) 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 RC series circuit

The impedance phase angle \({\theta}\) of the RC series circuit can be obtained from the vector diagram.

The impedance phase angle \({\theta}\) of the RC series circuit is expressed by the following equation:

\begin{eqnarray}

{\tan}{\theta}&=&\displaystyle\frac{-\displaystyle\frac{1}{{\omega}C}}{R}\\

\\

{\Leftrightarrow}{\theta}&=&{\tan}^{-1}\left(-\frac{1}{{\omega}CR}\right)\\

\end{eqnarray}

#### Summary

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

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

Thank you for reading.

Related article

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