LC Series Circuit (Impedance, Phasor Diagram)

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

• Equation, magnitude, vector diagram, and impedance phase angle of LC series circuit impedance

Impedance of the LC series circuit

An LC series circuit (also known as an LC filter or LC network) is an electrical circuit consisting of an inductor $$L$$ and a capacitor $$C$$ connected in series, 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\\
\\
{\dot{Z}}_C&=&-jX_C=-j\frac{1}{{\omega}C}=\frac{1}{j{\omega}C}
\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 $$C$$.

The impedance $${\dot{Z}}$$ of the LC series circuit is the sum of the respective impedance, and is as follow:

\begin{eqnarray}
{\dot{Z}}&=&{\dot{Z}}_L+{\dot{Z}}_C\\
\\
&=&jX_L-jX_C\\
\\
&=&j\left(X_L-X_C\right)\\
\\
&=&j\left({\omega}L-\frac{1}{{\omega}C}\right)
\end{eqnarray}

The impedance $${\dot{Z}}$$ can be divided into the following three cases, depending on the size of $$X_L$$ and $$X_C$$.

• In Case $$X_L{\;}{\gt}{\;}X_C$$
• The impedance $${\dot{Z}}$$ is positive($${\dot{Z}}{>}0$$) and inductive.
• In Case $$X_L{\;}{\lt}{\;}X_C$$
• The impedance $${\dot{Z}}$$ is negative($${\dot{Z}}{<}0$$) and capacitive.
• In Case $$X_L=X_C$$
• The impedance $${\dot{Z}}$$ is zero($${\dot{Z}}=0$$). In this case, the circuit is in series resonance. When series resonance is established, the angular frequency $${\omega}$$ and frequency $$f$$ are as follows:
\begin{eqnarray}
X_L&=&X_C\\
\\
{\omega}L&=&\frac{1}{{\omega}C}\\
\\
{\Leftrightarrow}{\omega}&=&\frac{1}{\sqrt{LC}}\\
\\
{\Leftrightarrow}f&=&\frac{1}{2{\pi}\sqrt{LC}}\\
\end{eqnarray}

Magnitude of the impedance of the LC series circuit

The magnitude $$Z$$ of the impedance $${\dot{Z}}$$ of the LC series circuit is the absolute value of  "$${\dot{Z}}=j\left({\omega}L-\displaystyle\frac{1}{{\omega}C}\right)$$".

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

\begin{eqnarray}
Z=|{\dot{Z}}|=\sqrt{\left({\omega}L-\displaystyle\frac{1}{{\omega}C}\right)^2}=\left|{\omega}L-\displaystyle\frac{1}{{\omega}C}\right|=\left|X_L-X_C\right|
\end{eqnarray}

The magnitude $$Z_L$$ of the impedance of the inductor $$L$$ and the magnitude $$Z_C$$ of the impedance of the capacitor $$C$$ are expressed as follows:

\begin{eqnarray}
Z_L&=&|{\dot{Z}}_L|=\sqrt{({\omega}L)^2}={\omega}L\\
\\
Z_C&=&|{\dot{Z}}_C|=\sqrt{\left(\displaystyle\frac{1}{{\omega}C}\right)^2}=\displaystyle\frac{1}{{\omega}C}
\end{eqnarray}

Supplement

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

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

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

Vector diagram of the LC series circuit

The vector diagram of the impedance $${\dot{Z}}$$ of the LC series circuit can be drawn in the following steps.

How to draw a Vector Diagram

• Draw a vector of impedance $${\dot{Z}}_L$$ of inductor $$L$$
• Draw a vector of impedance $${\dot{Z}}_C$$ of capacitor $$C$$
• Combine the vectors

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

Draw a vector of impedance $${\dot{Z}}_L$$ of inductor $$L$$

The impedance $${\dot{Z}}_L$$ of the inductor $$L$$ is expressed as "$${\dot{Z}}_L=j{\omega}L$$".

Therefore, the orientation of the impedance $${\dot{Z}}_L$$ vector is 90° counterclockwise around the real axis (with "$$+j$$", it rotates 90° counterclockwise). How to determine the vector orientation will be explained in detail later.

The magnitude (length) $$Z_L$$ of the vector of the impedance $${\dot{Z}}_L$$ is "$$Z_L=|{\dot{Z}}_L|={\omega}L$$".

Draw a vector of impedance $${\dot{Z}}_C$$ of capacitor $$C$$

The impedance $${\dot{Z}}_C$$ of the capacitor $$C$$ is expressed as "$${\dot{Z}}_C=-j\displaystyle\frac{1}{{\omega}C}$$".

Therefore, the orientation of the impedance $${\dot{Z}}_C$$ vector is 90° clockwise around the real axis (with "$$-j$$", it rotates 90° clockwise). How to determine the vector orientation will be explained in detail later.

The magnitude (length) $$Z_C$$ of the vector of the impedance $${\dot{Z}}_C$$ is "$$Z_C=|{\dot{Z}}_C|=\displaystyle\frac{1}{{\omega}C}$$".

Combine the vectors

Combining the vector of "impedance $${\dot{Z}}_L$$ of inductor $$L$$" and "impedance $${\dot{Z}}_C$$ of capacitor $$C$$" is the vector diagram of the impedance $${\dot{Z}}$$ of the LC series circuit.

The impedance $${\dot{Z}}$$ of the LC series circuit is the sum of the respective impedance, and is as follow:

\begin{eqnarray}
{\dot{Z}}&=&j\left(X_L-X_C\right)\\
\\
&=&j\left({\omega}L-\frac{1}{{\omega}C}\right)
\end{eqnarray}

The magnitude of $$X_L$$ and $$X_C$$ in the parentheses in the above equation changes the vector direction of the impedance $${\dot{Z}}$$.

• In Case $$X_L{\;}{\gt}{\;}X_C$$
• The vector direction of the impedance $${\dot{Z}}$$ is upward.
• In Case $$X_L{\;}{\lt}{\;}X_C$$
• The vector direction of the impedance $${\dot{Z}}$$ is downward.
• In Case $$X_L=X_C$$
• The impedance $${\dot{Z}}$$ is "zero". Therefore, there is no vector.

The magnitude (length) $$Z$$ of the vector of the impedance $${\dot{Z}}$$ can be expressed as follows.

\begin{eqnarray}
Z=\left|{\omega}L-\displaystyle\frac{1}{{\omega}C}\right|=\left|X_L-X_C\right|
\end{eqnarray}

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 impedance $${\dot{Z}}_L$$ of inductor $$L$$ is represented by "$${\dot{Z}}_L=j{\omega}L$$". Therefore, the direction of vector $${\dot{Z}}_L$$ is 90° counterclockwise around the real axis.

The impedance $${\dot{Z}}_C$$ of capacitor $$C$$ is represented by "$${\dot{Z}}_C=-j\displaystyle\frac{1}{{\omega}C}$$". Therefore, the direction of vector $${\dot{Z}}_C$$ is 90° clockwise around the real axis.

Impedance phase angle of the LC series circuit

The impedance phase angle ({\theta}) varies depending on the size of $$X_L$$ and $$X_C$$.

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