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}

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.

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

Vector orientation

Here is a more detailed explanation of how vector orientation is determined.

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

Thank you for reading.