RL Parallel Circuit (Admittance, Phasor Diagram)

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

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

Admittance of the RL parallel circuit

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

The impedance $${\dot{Z}}_R$$ of the resistor $$R$$ and the impedance $${\dot{Z}}_L$$ of the inductor $$L$$ 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}
\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 $$L$$.

Since admittance is the reciprocal of impedance, the admittance $${\dot{Y}}_R$$ of resistor $$R$$ and the admittance $${\dot{Y}}_L$$ of inductor $$L$$ 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}_L}&=&\frac{1}{{\dot{Z}_L}}=\frac{1}{j{\omega}L}=-j\frac{1}{{\omega}L}\tag{4}
\end{eqnarray}

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

\begin{eqnarray}
{\dot{Y}}&=&{\dot{Y}_R}+{\dot{Y}_L}\\
\\
&=&\frac{1}{R}+\frac{1}{j{\omega}L}\\
\\
&=&\frac{1}{R}-j\frac{1}{{\omega}L}\tag{5}
\end{eqnarray}

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

Admittance of the RL parallel circuit

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

Magnitude of the admittance of the RL parallel circuit

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

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

The magnitude $$Y$$ of the admittance of the RL 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 $$\displaystyle\frac{1}{{\omega}L}$$ 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(\displaystyle\frac{1}{{\omega}L}\right)^2}\\
\\
&=&\displaystyle\sqrt{\displaystyle\frac{{\omega}^2L^2+R^2}{{\omega}^2L^2R^2}}\\
\\
&=&\displaystyle\frac{\displaystyle\sqrt{R^2+{\omega}^2L^2}}{{\omega}LR}\tag{8}
\end{eqnarray}

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

Magnitude of the admittance of the RL parallel circuit

\begin{eqnarray}
Y=|{\dot{Y}}|=\displaystyle\sqrt{\left(\displaystyle\frac{1}{R}\right)^2+\left(\displaystyle\frac{1}{{\omega}L}\right)^2}=\displaystyle\frac{\displaystyle\sqrt{R^2+{\omega}^2L^2}}{{\omega}LR}{\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 RL parallel circuit

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

Combine the vectors

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

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

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

Supplement

The magnitude (length) $$Y=\displaystyle\sqrt{\left(\displaystyle\frac{1}{R}\right)^2+\left(\displaystyle\frac{1}{{\omega}L}\right)^2}$$ of the vector of the synthetic admittance $${\dot{Y}}$$ of the RL 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 $${\dot{Y}_L}$$ of the inductor $$L$$ is expressed by the following equation

\begin{eqnarray}
{\dot{Y}_L}=-j\frac{1}{{\omega}L}\tag{16}
\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 RL parallel circuit

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

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

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

Admittance phase angle of the RL parallel circuit

\begin{eqnarray}
{\theta}={\tan}^{-1}\left(-\displaystyle\frac{R}{{\omega}L}\right)\tag{18}
\end{eqnarray}

Summary

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

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

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