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

- Equation, magnitude, vector diagram, and admittance phase angle of LC parallel circuit admittance

## Admittance of the LC parallel circuit

An LC parallel circuit is an electrical circuit consisting of an inductor \(L\) and a capacitor \(C\) connected in parallel, driven by a voltage source or current source.

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}}_L&=&jX_L=j{\omega}L\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 \(X_L\left(={\omega}L\right)\) 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 \({\dot{Y}}_L\) of inductor \(L\) and the admittance \({\dot{Y}}_C\) of capacitor \(C\) can each be expressed by the following equations.

\begin{eqnarray}

{\dot{Y}_L}&=&\frac{1}{{\dot{Z}_L}}=\frac{1}{j{\omega}L}=-j\frac{1}{{\omega}L}\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 LC parallel circuit is the sum of the respective admittances, and is as follows:

\begin{eqnarray}

{\dot{Y}}&=&{\dot{Y}_L}+{\dot{Y}_C}\\

\\

&=&\frac{1}{j{\omega}L}+j{\omega}C\\

\\

&=&-j\frac{1}{{\omega}L}+j{\omega}C\\

\\

&=&j\left({\omega}C-\frac{1}{{\omega}L}\right)\\

\\

&=&j\left(\frac{1}{X_C}-\frac{1}{X_L}\right)\tag{5}

\end{eqnarray}

From the above, the admittance \({\dot{Y}}\) of the LC parallel circuit becomes the following equation.

Admittance of the LC parallel circuit

\begin{eqnarray}

{\dot{Y}}&=&j\left({\omega}C-\frac{1}{{\omega}L}\right){\mathrm{[S]}}\\

\\

&=&j\left(\frac{1}{X_C}-\frac{1}{X_L}\right){\mathrm{[S]}}\tag{6}

\end{eqnarray}

### Magnitude of the admittance of the LC parallel circuit

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

\begin{eqnarray}

{\dot{Y}}&=&j\left({\omega}C-\frac{1}{{\omega}L}\right){\mathrm{[S]}}\\

\\

&=&j\left(\frac{1}{X_C}-\frac{1}{X_L}\right){\mathrm{[S]}}\tag{7}

\end{eqnarray}

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

In more detail, the magnitude \(Y\) of the admittance \({\dot{Y}}\) is obtained by taking the square root of the square of the imaginary part \(\left({\omega}C-\displaystyle\frac{1}{{\omega}L}\right)\), which can be expressed in the following equation.

Magnitude of the admittance of the LC parallel circuit

\begin{eqnarray}

Y&=&|{\dot{Y}}|=\sqrt{\left({\omega}C-\frac{1}{{\omega}L}\right)^2}=\left|{\omega}C-\frac{1}{{\omega}L}\right|=\left|\frac{1}{X_C}-\frac{1}{X_L}\right|{\mathrm{[S]}}\tag{8}

\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 LC parallel circuit

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

How to draw a Vector Diagram

- 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 \({\dot{Y}}_L\) 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{9}

\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{10}

\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{11}

\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{12}

\end{eqnarray}

### Combine the vectors

Combining the vector of "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 LC parallel circuit.

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

\begin{eqnarray}

{\dot{Y}}&=&j\left({\omega}C-\frac{1}{{\omega}L}\right)=j\left(\frac{1}{X_C}-\frac{1}{X_L}\right)=j\left(Y_C-Y_L\right){\mathrm{[S]}}\tag{13}\\

\\

Y&=&\left|{\omega}C-\frac{1}{{\omega}L}\right|=\left|\frac{1}{X_C}-\frac{1}{X_L}\right|=\left|Y_C-Y_L\right|{\mathrm{[S]}}\tag{14}

\end{eqnarray}

In the above equation, the vector direction of admittance \({\dot{Y}}\) of the LC 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{15}

\end{eqnarray}

The vector direction of admittance \({\dot{Y}}\) of the LC parallel circuit is upward, 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{16}

\end{eqnarray}

The vector direction of admittance \({\dot{Y}}\) of the LC parallel circuit is downward, 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{17}

\end{eqnarray}

At this time, the admittance \({\dot{Y}}\) of the LC parallel circuit becomes the following equation.

\begin{eqnarray}

{\dot{Y}}&=&j\left({\omega}C-\frac{1}{{\omega}L}\right)\\

\\

&=&j\left(Y_C-Y_L\right)\\

\\

&=&j\left(0\right)\\

\\

&=&0\tag{18}

\end{eqnarray}

The admittance \({\dot{Y}}\) is "zero". Therefore, there is no vector.

### 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{19}

\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{20}

\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 LC parallel circuit

The admittance phase angle \({\theta}\) of the LC parallel circuit depends on the relationship between "inductive reactance \(X_L(={\omega}L)\)" and "capacitive reactance \(X_C\left(=\displaystyle\frac{1}{{\omega}C}\right)\)".

- In Case \(X_L{\;}{\gt}{\;}X_C\)
- The admittance phase angle \({\theta}\) is the following value:

\begin{eqnarray}

{\theta}=\frac{{\pi}}{2}{\mathrm{[rad]}}

\end{eqnarray}

- The admittance phase angle \({\theta}\) is the following value:
- In Case \(X_L{\;}{\lt}{\;}X_C\)
- The admittance phase angle \({\theta}\) is the following value:

\begin{eqnarray}

{\theta}=-\frac{{\pi}}{2}{\mathrm{[rad]}}

\end{eqnarray}

- The admittance phase angle \({\theta}\) is the following value:
- In Case \(X_L=X_C\)
- The admittance phase angle \({\theta}\) is the following value:

\begin{eqnarray}

{\theta}=0{\mathrm{[rad]}}

\end{eqnarray}

- The admittance phase angle \({\theta}\) is the following value:

#### Summary

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

- Equation, magnitude, vector diagram, and admittance phase angle of LC parallel circuit admittance

Thank you for reading.

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