# LC Parallel Circuit (Admittance, Phasor Diagram)

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}
\end{eqnarray}
• In Case $$X_L{\;}{\lt}{\;}X_C$$
• The admittance phase angle $${\theta}$$ is the following value:
\begin{eqnarray}
\end{eqnarray}
• In Case $$X_L=X_C$$
• The admittance phase angle $${\theta}$$ is the following value:
\begin{eqnarray}