Cutoff Frequency of a Bandpass Filter

The Band Pass Filter consists of two cutoff frequencies. The bandpass filter is made of a high pass and a low pass filter. The first cutoff frequency is from a high pass filter, known as the higher cutoff frequency. This cut-off frequency is known as FC high.

Cutoff Frequency of Band Pass Filter — Cut off Frequency of a Bandpass Filter

$\begin{align*}FC_{high}=\frac{1}{2\pi R_{1}C_{1}}\end{align*}$

The second cutoff frequency is from the low pass filter known as the lower cutoff frequency. This cut-off frequency is known as FC low.

$\begin{align*}FC_{low}=\frac{1}{2\pi R_{2}C_{2}}\end{align*}$

Bandwidth is given as the range between these frequencies. For a high pass filter, the cut-off frequency will define the lower value of bandwidth. For a low pass filter, the cutoff frequency will define the higher value of bandwidth.

Cut off Frequency of RL circuit

Consider a simple RL circuit as shown below.

The transfer function for the same is given as

$\begin{align*}\frac{V_{0}(s)}{V_{i}(s)}=\frac{R}{sL+R}\end{align*}$

$\begin{align*}H(s)=\frac{\frac{R}{L}}{S+\frac{R}{L}}\end{align*}$

Substitute $s=j\omega$ in the above equation to calculate the frequency response

$\begin{align*}H(j\omega)=\frac{\frac{R}{L}}{j\omega+\frac{R}{L}}\end{align*}$

Magnitude Response is

$\begin{align*}\left |H(j\omega) \right |=\frac{\frac{R}{L}}{\sqrt{\omega^{2}+\left ( \frac{R}{L}\right)^{2}}}\end{align*}$

When $\omega$ = 0

$\begin{align*}\left |H(j0) \right |=\frac{\frac{R}{L}}{\sqrt{0^{2}+\left ( \frac{R}{L}\right)^{2}}}=1\end{align*}$

When $\omega$ = $\infty$

$\begin{align*}\left |H(j\infty) \right |=\frac{\frac{R}{L}}{\sqrt{\infty^{2}+\left ( \frac{R}{L}\right)^{2}}}=0\end{align*}$

To calculate the cutoff frequency,

$\begin{align*}\left |H(j\omega_c) \right |=\frac{\frac{R}{L}}{\sqrt{\omega_c^{2}+\left ( \frac{R}{L}\right)^{2}}}=\frac{{1}}{\sqrt{2}} \end{align*}$

Finally, cut off frequency of an RL circuit is given as

$\begin{align*} \omega_{c}=\frac{R}{L} \end{align*}$

Cutoff Frequency of RL Circuit — Cutoff Frequency of an RL Circuit

Cut off Frequency of RC circuit

Consider a simple RC circuit as shown below.

The transfer function for the same is given as

$\begin{align*}\frac{V_{0}(s)}{V_{i}(s)}=\frac{\frac{1}{sC}}{R+\frac{1}{sC}}\end{align*}$

$\begin{align*}H(s)=\frac{\frac{1}{RC}}{S+\frac{1}{RC}}\end{align*}$

Substitute $s=j\omega$ in the above equation to calculate the frequency response

$\begin{align*}H(j\omega)=\frac{\frac{1}{RC}}{j\omega+\frac{1}{RC}}\end{align*}$

Magnitude Response is

$\begin{align*}\left |H(j\omega) \right |=\frac{\frac{1}{RC}}{\sqrt{\omega^{2}+\left ( \frac{1}{RC}\right)^{2}}}\end{align*}$

When $\omega$ = 0

$\begin{align*}\left |H(j0) \right |=\frac{\frac{1}{RC}}{\sqrt{0^{2}+\left ( \frac{1}{RC}\right)^{2}}}=1\end{align*}$

When $\omega$ = $\infty$

$\begin{align*}\left |H(j\infty) \right |=\frac{\frac{1}{RC}}{\sqrt{\infty^{2}+\left ( \frac{1}{RC}\right)^{2}}}=0\end{align*}$

To calculate the cutoff frequency,

$\begin{align*}\left |H(j\omega_c) \right |=\frac{\frac{1}{RC}}{\sqrt{\omega_c^{2}+\left ( \frac{1}{RC}\right)^{2}}}=\frac{{1}}{\sqrt{2}} \end{align*}$

Finally, cut off frequency of an RL circuit is given as

$\begin{align*} \omega_{c}=\frac{1}{RC} \end{align*}$

Cutoff Frequency Of RC Circuit — Cutoff Frequency of RC Circuit

Cut off Frequency of RL circuit

Cut off Frequency of RC circuit

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