# RC Parallel Circuit (Admittance, Phasor Diagram)

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

• Equation, magnitude, vector diagram, and admittance phase angle of RC parallel circuit admittance

## Admittance of the RC parallel circuit

An RC parallel circuit is an electrical circuit consisting of a resistor $$R$$ and a capacitor $$C$$ connected in parallel, 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\tag{1}\\
\\
{\dot{Z}_C}&=&-jX_C=-j\frac{1}{{\omega}C}=\frac{1}{j{\omega}C}\tag{2}
\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 $$C$$.

Since admittance is the reciprocal of impedance, the admittance $${\dot{Y}}_R$$ of resistor $$R$$ and the admittance $${\dot{Y}}_C$$ of capacitor $$C$$ can each be expressed by the following equations.

\begin{eqnarray}
{\dot{Y}_R}&=&\frac{1}{{\dot{Z}_R}}=\frac{1}{R}\tag{3}\\
\\
{\dot{Y}_C}&=&\frac{1}{{\dot{Z}_C}}=\frac{1}{\displaystyle\frac{1}{j{\omega}C}}=j{\omega}C\tag{4}
\end{eqnarray}

The admittance $${\dot{Y}}$$ of the RC parallel circuit is the sum of the respective admittances, and is as follows:

\begin{eqnarray}
{\dot{Y}}&=&{\dot{Y}_R}+{\dot{Y}_C}\\
\\
&=&\frac{1}{R}+j{\omega}C\tag{5}
\end{eqnarray}

From the above, the admittance $${\dot{Y}}$$ of the RC parallel circuit becomes the following equation.

Admittance of the RC parallel circuit

\begin{eqnarray}
{\dot{Y}}=\frac{1}{R}+j{\omega}C{\mathrm{[S]}}\tag{6}
\end{eqnarray}

### Magnitude of the admittance of the RC parallel circuit

We have just obtained the admittance $${\dot{Y}}$$ expressed by the following equation.

\begin{eqnarray}
{\dot{Y}}=\frac{1}{R}+j{\omega}C{\mathrm{[S]}}\tag{7}
\end{eqnarray}

The magnitude $$Y$$ of the admittance of the RC parallel circuit is the absolute value of the admittance $${\dot{Y}}$$ in equation (7).

In more detail, the magnitude $$Y$$ of the admittance $${\dot{Y}}$$ can be obtained by adding the square of the real part $$\displaystyle\frac{1}{R}$$ and the square of the imaginary part $${\omega}C$$ and taking the square root, which can be expressed in the following equation.

\begin{eqnarray}
Y&=&|{\dot{Y}}|\\
\\
&=&\displaystyle\sqrt{\left(\displaystyle\frac{1}{R}\right)^2+\left({\omega}C\right)^2}\\
\\
&=&\displaystyle\sqrt{\displaystyle\frac{1+{\omega}^2C^2R^2}{R^2}}\\
\\
&=&\displaystyle\frac{\displaystyle\sqrt{1+{\omega}^2C^2R^2}}{R}\tag{8}
\end{eqnarray}

From the above, the magnitude $$Y$$ of the admittance $${\dot{Y}}$$ of the RC parallel circuit becomes the following equation.

Magnitude of the admittance of the RC parallel circuit

\begin{eqnarray}
Y=|{\dot{Y}}|=\displaystyle\sqrt{\left(\displaystyle\frac{1}{R}\right)^2+\left({\omega}C\right)^2}=\displaystyle\frac{\displaystyle\sqrt{1+{\omega}^2C^2R^2}}{R}{\mathrm{[S]}}\tag{9}
\end{eqnarray}

Supplement

Some admittance $$Y$$ symbols have a ". (dot)" above them and are labeled $${\dot{Y}}$$.

$${\dot{Y}}$$ with this dot represents a vector.

If it has a dot (e.g. $${\dot{Y}}$$), it represents a vector (complex number), and if it does not have a dot (e.g. $$Y$$), it represents the absolute value (magnitude, length) of the vector.

## Vector diagram of the RC parallel circuit

The vector diagram of the admittance $${\dot{Y}}$$ of the RC parallel circuit can be drawn in the following steps.

How to draw a Vector Diagram

• Draw a vector of admittance $${\dot{Y}}_R$$ of resistor $$R$$
• Draw a vector of admittance $${\dot{Y}}_C$$ of capacitor $$C$$
• Combine the vectors

Let's take a look at each step in turn.

### Draw a vector of admittance $${\dot{Y}}_R$$ of resistor $$R$$

The admittance $${\dot{Y}}_R$$ of resistor $$R$$ is expressed by the following equation.

\begin{eqnarray}
{\dot{Y}_R}=\frac{1}{R}\tag{10}
\end{eqnarray}

Therefore, the vector direction of the admittance $${\dot{Y}}_R$$ is the direction of the real axis. If the expression does not have an imaginary unit $$j$$, the vector does not rotate and is oriented on the real axis. How to determine the vector orientation will be explained in more detail later.

Also, the magnitude (length) $$Y_R$$ of the vector of the admittance $${\dot{Y}}_R$$ is expressed by the following equation.

\begin{eqnarray}
Y_R=|{\dot{Y}_R}|=\displaystyle\sqrt{\left(\frac{1}{R}\right)^2}=\frac{1}{R}\tag{11}
\end{eqnarray}

### Draw a vector of admittance $${\dot{Y}}_C$$ of capacitor $$C$$

The admittance $${\dot{Y}}_C$$ of capacitor $$C$$ is expressed by the following equation.

\begin{eqnarray}
{\dot{Y}_C}=j{\omega}C\tag{12}
\end{eqnarray}

Therefore, the orientation of the admittance $${\dot{Y}}_C$$ vector is 90° counterclockwise around the real axis. When "$$+j$$" is added to the equation, the vector is rotated 90° counterclockwise. How to determine the vector orientation will be explained in detail later.

Also, the magnitude (length) $$Y_C$$ of the vector of the admittance $${\dot{Y}}_C$$ is expressed by the following equation.

\begin{eqnarray}
Y_C=|{\dot{Y}_C}|=\displaystyle\sqrt{\left({\omega}C\right)^2}={\omega}C=\frac{1}{X_C}\tag{13}
\end{eqnarray}

### Combine the vectors

Combining the vector of "admittance $${\dot{Y}}_R$$ of resistor $$R$$" and "admittance $${\dot{Y}}_C$$ of capacitor $$C$$" is the vector diagram of the admittance $${\dot{Y}}$$ of the RC parallel circuit.

We have just explained that the admittance $${\dot{Y}}$$ of an RC parallel circuit and its magnitude $$Y$$ are expressed by the following equation

\begin{eqnarray}
{\dot{Y}}&=&\frac{1}{R}+j{\omega}C\tag{14}\\
\\
Y&=&\displaystyle\sqrt{\left(\displaystyle\frac{1}{R}\right)^2+\left({\omega}C\right)^2}\tag{15}
\end{eqnarray}

Supplement

The magnitude (length) $$Y=\displaystyle\sqrt{\left(\displaystyle\frac{1}{R}\right)^2+\left({\omega}C\right)^2}$$ of the vector of the synthetic admittance $${\dot{Y}}$$ of the RC parallel 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 admittance $${\dot{Y}_C}$$ of the capacitor $$C$$ is expressed by the following equation

\begin{eqnarray}
{\dot{Y}_C}=j{\omega}C\tag{16}
\end{eqnarray}

Since the expression for the admittance $${\dot{Y}_C}$$ of the capacitor $$C$$ has '$$+j$$', the direction of the vector $${\dot{Y}_C}$$ is 90° counterclockwise rotation around the real axis.

## Admittance phase angle of the RC parallel circuit

The admittance phase angle $${\theta}$$ of the RC parallel circuit can be obtained from the vector diagram.

\begin{eqnarray}
{\tan}{\theta}&=&\displaystyle\frac{{\omega}C}{\displaystyle\frac{1}{R}}\\
\\
&=&{\omega}CR\\
\\
{\Leftrightarrow}{\theta}&=&{\tan}^{-1}\left({\omega}CR\right)\tag{17}
\end{eqnarray}

From the above, the admittance phase angle $${\theta}$$ of the RC parallel circuit is expressed by the following equation.

Admittance phase angle of the RC parallel circuit

\begin{eqnarray}
{\theta}={\tan}^{-1}\left({\omega}CR\right)\tag{18}
\end{eqnarray}

#### Summary

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

1. Equation, magnitude, vector diagram, and admittance phase angle of RC parallel circuit admittance