# RC Series Circuit (Impedance, Phasor Diagram)

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.

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