# RLC Parallel Circuit (Admittance, Phasor Diagram)

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

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

## Admittance of the RLC parallel circuit

An RLC parallel circuit is an electrical circuit consisting of a resistor $$R$$, an inductor $$L$$, and a capacitor $$C$$ connected in parallel, driven by a voltage source or current source.

The impedance $${\dot{Z}}_R$$ of the resistor $$R$$, the impedance $${\dot{Z}}_L$$ of the inductor $$L$$, 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}_L}&=&jX_L=j{\omega}L\tag{2}\\
\\
{\dot{Z}_C}&=&-jX_C=-j\frac{1}{{\omega}C}=\frac{1}{j{\omega}C}\tag{3}
\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 $$L$$ 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 $$R$$, the admittance $${\dot{Y}}_L$$ of inductor $$L$$, 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{4}\\
\\
{\dot{Y}_L}&=&\frac{1}{{\dot{Z}_L}}=\frac{1}{j{\omega}L}=-j\frac{1}{{\omega}L}\tag{5}\\
\\
{\dot{Y}_C}&=&\frac{1}{{\dot{Z}_C}}=\frac{1}{\displaystyle\frac{1}{j{\omega}C}}=j{\omega}C\tag{6}
\end{eqnarray}

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

\begin{eqnarray}
{\dot{Y}}&=&{\dot{Y}_R}+{\dot{Y}_L}+{\dot{Y}_C}\\
\\
&=&\frac{1}{R}+\frac{1}{j{\omega}L}+j{\omega}C\\
\\
&=&\frac{1}{R}-j\frac{1}{{\omega}L}+j{\omega}C\\
\\
&=&\frac{1}{R}+j\left({\omega}C-\frac{1}{{\omega}L}\right)\\
\\
&=&\frac{1}{R}+j\left(\frac{1}{X_C}-\frac{1}{X_L}\right)\tag{7}
\end{eqnarray}

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

Admittance of the RLC parallel circuit

\begin{eqnarray}
{\dot{Y}}&=&\frac{1}{R}+j\left({\omega}C-\frac{1}{{\omega}L}\right){\mathrm{[S]}}\\
\\
&=&\frac{1}{R}+j\left(\frac{1}{X_C}-\frac{1}{X_L}\right){\mathrm{[S]}}\tag{8}
\end{eqnarray}

### Magnitude of the admittance of the RLC parallel circuit

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

\begin{eqnarray}
{\dot{Y}}&=&\frac{1}{R}+j\left({\omega}C-\frac{1}{{\omega}L}\right){\mathrm{[S]}}\\
\\
&=&\frac{1}{R}+j\left(\frac{1}{X_C}-\frac{1}{X_L}\right){\mathrm{[S]}}\tag{9}
\end{eqnarray}

The magnitude $$Y$$ of the admittance of the RLC parallel circuit is the absolute value of the admittance $${\dot{Y}}$$ in equation (9).

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

\begin{eqnarray}
Y&=&|{\dot{Y}}|\\
\\
&=&\sqrt{\left(\frac{1}{R}\right)^2+\left(\frac{1}{X_C}-\frac{1}{X_L}\right)^2}\\
\\
&=&\sqrt{\left(\frac{1}{R}\right)^2+\left({\omega}C-\frac{1}{{\omega}L}\right)^2}\\
\\
&=&\sqrt{\frac{\left(\displaystyle\frac{1}{R}\right)^2×{\omega}^2L^2R^2+\left({\omega}C-\displaystyle\frac{1}{{\omega}L}\right)^2×{\omega}^2L^2R^2}{{\omega}^2L^2R^2}}\\
\\
&=&\sqrt{\frac{{\omega}^2L^2+\left({\omega}^2LC-1\right)^2×R^2}{{\omega}^2L^2R^2}}\\
\\
&=&\frac{\sqrt{{\omega}^2L^2+R^2\left({\omega}^2LC-1\right)^2}}{{\omega}LR}\tag{10}
\end{eqnarray}

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

Magnitude of the admittance of the RLC parallel circuit

\begin{eqnarray}
Y&=&|{\dot{Y}}|\\
\\
&=&\sqrt{\left(\frac{1}{R}\right)^2+\left(\frac{1}{X_C}-\frac{1}{X_L}\right)^2}{\mathrm{[S]}}\\
\\
&=&\sqrt{\left(\frac{1}{R}\right)^2+\left({\omega}C-\frac{1}{{\omega}L}\right)^2}{\mathrm{[S]}}\\
\\
&=&\frac{\sqrt{{\omega}^2L^2+R^2\left({\omega}^2LC-1\right)^2}}{{\omega}LR}{\mathrm{[S]}}\tag{11}
\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 RLC parallel circuit

The vector diagram of the admittance $${\dot{Y}}$$ of the RLC 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 $${\dot{Y}}_L$$ of inductor $$L$$
• 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{12}
\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{13}
\end{eqnarray}

### Draw a vector of admittance $${\dot{Y}}_L$$ of inductor $$L$$

The admittance $${\dot{Y}}_L$$ of inductor $$L$$ is expressed by the following equation.

\begin{eqnarray}
{\dot{Y}_L}=-j\frac{1}{{\omega}L}\tag{14}
\end{eqnarray}

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

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

\begin{eqnarray}
Y_L=|{\dot{Y}_L}|=\displaystyle\sqrt{\left(\frac{1}{{\omega}L}\right)^2}=\frac{1}{{\omega}L}=\frac{1}{X_L}\tag{15}
\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{16}
\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{17}
\end{eqnarray}

### Combine the vectors

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

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

\begin{eqnarray}
{\dot{Y}}&=&\frac{1}{R}+j\left({\omega}C-\frac{1}{{\omega}L}\right){\mathrm{[S]}}\tag{18}\\
\\
Y&=&\sqrt{\left(\frac{1}{R}\right)^2+\left({\omega}C-\frac{1}{{\omega}L}\right)^2}{\mathrm{[S]}}\tag{19}
\end{eqnarray}

In the above equation, the vector direction of admittance $${\dot{Y}}$$ of the RLC parallel circuit changes depending on the magnitude of "inductive reactance $$X_L={\omega}L$$" and "capacitive reactance $$X_C=\displaystyle\frac{1}{{\omega}C}$$".

• In Case $$X_L{\;}{\gt}{\;}X_C$$
• In Case $$X_L{\;}{\lt}{\;}X_C$$
• In Case $$X_L=X_C$$

In Case $$X_L{\;}{\gt}{\;}X_C$$

If "inductive reactance $$X_L$$" is larger than "capacitive reactance $$X_C$$", the following equation holds.

\begin{eqnarray}
&&X_L{\;}{\gt}{\;}X_C\\
\\
{\Leftrightarrow}&&\frac{1}{Y_L}{\;}{\gt}{\;}\frac{1}{Y_C}\\
\\
{\Leftrightarrow}&&Y_L{\;}{\lt}{\;}Y_C\tag{20}
\end{eqnarray}

The vector direction of admittance $${\dot{Y}}$$ of the RLC parallel circuit is upward to the right, because "the magnitude $$Y_L$$ of admittance of inductor $$L$$" is smaller than "the magnitude $$Y_C$$ of admittance of capacitor $$C$$".

In Case $$X_L{\;}{\lt}{\;}X_C$$

If "inductive reactance $$X_L$$" is smaller than "capacitive reactance $$X_C$$", the following equation holds.

\begin{eqnarray}
&&X_L{\;}{\lt}{\;}X_C\\
\\
{\Leftrightarrow}&&\frac{1}{Y_L}{\;}{\lt}{\;}\frac{1}{Y_C}\\
\\
{\Leftrightarrow}&&Y_L{\;}{\gt}{\;}Y_C\tag{21}
\end{eqnarray}

The vector direction of admittance $${\dot{Y}}$$ of the RLC parallel circuit is downward to the right, because "the magnitude $$Y_L$$ of admittance of inductor $$L$$" is larger than "the magnitude $$Y_C$$ of admittance of capacitor $$C$$".

In Case $$X_L=X_C$$

If "inductive reactance $$X_L$$" is equal to "capacitive reactance $$X_C$$", the following equation holds.

\begin{eqnarray}
&&X_L=X_C\\
\\
{\Leftrightarrow}&&\frac{1}{Y_L}=\frac{1}{Y_C}\\
\\
{\Leftrightarrow}&&Y_L=Y_C\\
\\
{\Leftrightarrow}&&Y_C-Y_L=0\tag{22}
\end{eqnarray}

At this time, the admittance $${\dot{Y}}$$ of the RLC parallel circuit becomes the following equation.

\begin{eqnarray}
{\dot{Y}}&=&\frac{1}{R}+j\left({\omega}C-\frac{1}{{\omega}L}\right)\\
\\
&=&\frac{1}{R}+j\left(Y_C-Y_L\right)\\
\\
&=&\frac{1}{R}+j\left(0\right)\\
\\
&=&\frac{1}{R}\tag{23}
\end{eqnarray}

Since the above equation does not have an imaginary unit $$j$$, the vector is not rotated. Therefore, the direction of the vector of admittance $${\dot{Y}}$$ is to the right.

Supplement

The magnitude (length) $$Y$$ of the vector of the synthetic admittance $${\dot{Y}}$$ of the RLC 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{24}
\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.

The admittance $${\dot{Y}_L}$$ of the inductor $$L$$ is expressed by the following equation

\begin{eqnarray}
{\dot{Y}_L}=-j\frac{1}{{\omega}L}\tag{25}
\end{eqnarray}

Since the expression for the admittance $${\dot{Y}_L}$$ of the inductor $$L$$ has '$$-j$$', the direction of the vector $${\dot{Y}_L}$$ is 90° clockwise rotation around the real axis.

## Admittance phase angle of the RLC parallel circuit

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

\begin{eqnarray}
{\tan}{\theta}&=&\displaystyle\frac{{\omega}C-\displaystyle\frac{1}{{\omega}L}}{\displaystyle\frac{1}{R}}\\
\\
&=&\displaystyle\frac{\displaystyle\frac{1}{X_C}-\displaystyle\frac{1}{X_L}}{\displaystyle\frac{1}{R}}\\
\\
{\Leftrightarrow}{\theta}&=&{\tan}^{-1}\left(\displaystyle\frac{{\omega}C-\displaystyle\frac{1}{{\omega}L}}{\displaystyle\frac{1}{R}}\right)\\
\\
\\&=&{\tan}^{-1}\left(\displaystyle\frac{\displaystyle\frac{1}{X_C}-\displaystyle\frac{1}{X_L}}{\displaystyle\frac{1}{R}}\right)\tag{26}
\end{eqnarray}

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

Admittance phase angle of the RLC parallel circuit

\begin{eqnarray}
{\theta}&=&{\tan}^{-1}\left(\displaystyle\frac{{\omega}C-\displaystyle\frac{1}{{\omega}L}}{\displaystyle\frac{1}{R}}\right)\\
\\
&=&{\tan}^{-1}\left(\displaystyle\frac{\displaystyle\frac{1}{X_C}-\displaystyle\frac{1}{X_L}}{\displaystyle\frac{1}{R}}\right)\tag{27}
\end{eqnarray}

The magnitude of the inductive reactance $$X_L(={\omega}L)$$ and capacitive reactance $$X_C\left(=\displaystyle\frac{1}{{\omega}C}\right)$$ determine whether the admittance phase angle $${\theta}$$ of the RLC parallel circuit is positive or negative.

• In Case $$X_L{\;}{\gt}{\;}X_C$$
• In Case $$X_L{\;}{\lt}{\;}X_C$$
• In Case $$X_L=X_C$$

In Case $$X_L{\;}{\gt}{\;}X_C$$

If "inductive reactance $$X_L$$" is larger than "capacitive reactance $$X_C$$", the following equation holds.

\begin{eqnarray}
&&X_L{\;}{\gt}{\;}X_C\\
\\
{\Leftrightarrow}&&\frac{1}{X_C}-\frac{1}{X_L}{\;}{\gt}{\;}0\tag{28}
\end{eqnarray}

Therefore, the admittance phase angle $${\theta}$$ of the RLC parallel circuit is "positive".

In Case $$X_L{\;}{\lt}{\;}X_C$$

If "inductive reactance $$X_L$$" is smaller than "capacitive reactance $$X_C$$", the following equation holds.

\begin{eqnarray}
&&X_L{\;}{\lt}{\;}X_C\\
\\
{\Leftrightarrow}&&\frac{1}{X_C}-\frac{1}{X_L}{\;}{\lt}{\;}0\tag{29}
\end{eqnarray}

Therefore, the admittance phase angle $${\theta}$$ of the RLC parallel circuit is "negative".

In Case $$X_L=X_C$$

If "inductive reactance $$X_L$$" is equal to "capacitive reactance $$X_C$$", the following equation holds.

\begin{eqnarray}
&&X_L=X_C\\
\\
{\Leftrightarrow}&&\frac{1}{X_C}-\frac{1}{X_L}=0\tag{30}
\end{eqnarray}

Therefore, the admittance phase angle $${\theta}$$ of the RLC parallel circuit is "$${\theta}=0{\mathrm{[rad]}}$$".

#### Summary

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

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