Notch Filter (Bandstop): What is it? (Circuit & Design)

Contents

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Key learnings:

Notch Filter Defined: A notch filter (also known as a bandstop filter or reject filter) is defined as a device that blocks certain frequencies while allowing others to pass.
Circuit Overview: The notch filter circuit combines low-pass and high-pass filters in a configuration that blocks specific frequency bands.
Design Principles: Designing a notch filter involves selecting appropriate values for resistors, capacitors, and inductors to achieve the desired frequency rejection.
Transfer Function Explained: The transfer function of a notch filter describes how it attenuates certain frequencies while maintaining others, crucial for understanding filter behavior.
Practical Applications: Notch filters are essential in electronics to remove interference, such as the 50/60 Hz noise in medical ECG readings, enhancing signal clarity.

What is a Notch Filter (Bandstop Filter)?

A notch filter (also known as a bandstop filter or reject filter) blocks a chosen frequency band and passes the frequencies above and below that band. A true notch is a narrow stop. A wider stop is usually called a band-reject or band-elimination filter.

A notch is a band-stop filter with a narrow stopband and two passbands. As in the band-pass case, a band-reject filter can be wide or narrow.

A wide stop is a band-reject filter. A narrow stop is a notch. The magnitude shape is the complement of a band-pass: pass, stop, then pass again.

Example: a stop from 100 MHz to 200 MHz passes DC to 100 MHz and everything above 200 MHz. It rejects 100 MHz to 200 MHz. That span is a wide band-reject, not a sharp audio or mains notch.

In symbols, the filter passes from DC up to the lower cut-off frequency(f_L) and again above the upper cut-off (f_H). It rejects the band BW = f_H-f_L.

A common signal processing techniques job is to pull a narrowband interferer out of noise. A notch can remove that tone while leaving the rest of the band energy in place.

Notch Filter Circuit

One wide band-stop build puts a low-pass and a high-pass in parallel, then sums them. A cascade of the same pair would be a band-pass. The figure below is that parallel idea. A twin-T notch is a different, narrower RC network.

The upper arm is a passive RC low pass filter T: two resistors (R₁ and R₂) and a capacitor (C₁). That path passes frequencies below its own cutoff. On a wide band-stop that cutoff sits at the lower stop edge, not at f_H.

The lower arm is a passive RC high-pass filter T: capacitors (C₂ and C₃) and resistor (R₃). That path passes frequencies above its cutoff. On a wide band-stop that cutoff sits at the upper stop edge, not at f_L. The two T networks together are often called a Twin-T. A classic Twin-T notch is a matched R-C-R and C-R-C pair that rejects one frequency, not a wide LPF-plus-HPF stop.

A typical block view and a notch frequency response are shown below.

Typical Notch Filter Configuration — Typical Configuration of a Notch Filter

Frequency Response of A Notch Filter — Frequency Response

Notch Filter Types

Common builds are listed below.

Active Notch Filter

An active notch can be a low-pass and a high-pass in parallel, then an op-amp summer, as in the figure below.

That active circuit has three parts: an active low pass filter (LPF) in parallel with an active high-pass filter (HPF), then a summing amplifier or a Op-Amp adder. The summer adds the two paths. The figure above is that arrangement.

The low-pass cutoff (lower stop edge) is

$\begin{align*} f_L = \frac{1}{2\pi R_1C_1} \end{align*}$

The high-pass cutoff (upper stop edge) is

$\begin{align*} f_H = \frac{1}{2\pi R_2C_2} \end{align*}$

Passive Notch Filter

A passive notch uses only resistors, inductors and capacitors. There is no op-amp gain stage.

The RC figure below is a passive low-pass T in parallel with a passive high-pass T.

The upper arm is the low-pass T. The lower arm is the high-pass T.

Optical Notch Filter

An optical notch filter blocks a chosen wavelength band and transmits the rest. Rejection is often stated as optical density.

A grating or prism can spread light by wavelength. An optical notch then blocks one of those bands, for example a laser line and passes the others.

Real coatings also lose some light at other wavelengths. A well-made optical notch still passes the wanted bands with only small extra loss.

They are used in laser-based Raman spectroscopy, laser fluorescence and other life-science optics.

RLC Notch Filter

An RLC notch uses the three passive elements R, L and C. One series-LC shunt form is shown below.

In that form the shunt is series L and C, and the output is taken across that series pair. At series resonance the shunt impedance is a minimum, so that frequency is rejected. The equivalent circuit is shown below.

The series L-C impedance is

$\begin{align*} \begin{split} Z = j(X_L - X_C) \,\, \Omega \\ |Z| = \sqrt{(X_L - X_C)^2} \,\, \Omega \end{split} \end{align*}$

The net impedance including R is

$\begin{align*} \begin{split} Z_T_o_t_a_l = R+j(X_L - X_C) \,\, \Omega \\ |Z| = \sqrt{R^2+(X_L - X_C)^2} \,\, \Omega \end{split} \end{align*}$

From KVL (Kirchhoff’s Voltage Law) on that loop, the stored voltage ratio is

$\begin{align*} \begin{split} V_0 = V_i * \frac{|Z|}{Z_T_o_t_a_l} \\ V_0 = V_i * \frac{(X_L-X_C)^2}{\sqrt{R^2+(X_L - X_C)^2}} \end{split} \end{align*}$

At series resonance the resonant frequency is

$\begin{align*} f_r = \frac{1}{2\pi \sqrt{LC}} \end{align*}$

The series Q-factor is

$\begin{align*} Q = \frac{\omega_r L}{R} \end{align*}$

Where $\omega_r$ is the resonant angular frequency, 2π fr. The 3 dB bandwidth is $\frac{f_r}{Q}$ , not ωr itself.

Then $f_L = (f_r - \frac{BW}{2}) Hz$ is the lower cut-off, and

$f_H = (f_r + \frac{BW}{2}) Hz$ is the upper cut-off. Those arithmetic-mean edges are a good fit when the stop is narrow. The curve below is the magnitude shape.

Notch Filter Characteristics — Characteristics of a Notch Filter

Another RLC notch arrangement is shown below.

RLC Notch Filter Circuit 2 — RLC Notch Filter Circuit

RF Notch Filter

An RF notch is a narrow stop used, for example, to reject a carrier while measuring amplifier distortion products.

An LC circuit wave-trap can reject one strong interferer so a nearby receiver can still hear weaker stations. That is the old broadcast “wave trap” use.

Strong FM broadcast carriers can also overload an SDR. An FM-band notch ahead of the SDR USB dongle is a common fix.

Butterworth Notch Filter

A Butterworth notch is a Butterworth prototype mapped to a stopband. Butterworth means a maximally flat passband, not a flat stopband.

An ideal notch would reject the stop and pass everything else with no extra distortion. A simple first-order trap is often too shallow when the signal-to-noise ratio is already low. A higher-order accuracy-minded Butterworth notch can make a deeper, better-shaped stop.

On an ECG (electrocardiogram) the usual extras are mains hum, instrument noise, external magnetic field interference, motion artefact and breathing. Each sits in a different frequency range.

Some of that noise sits inside the ECG band, so a blunt filter also hurts the wanted trace. A 4^th order Butterworth notch around 50 Hz (or 60 Hz) is a published way to cut mains hum. The “3 dB stopband” wording is kept as the page’s own spec; check the paper or datasheet for the exact edges.

Notch Filter Design Example

Design an RLC notch with cut-offs 23 kHz and 25 kHz. Take Inductance L = 45 mH.

Given: f_L=23 kHz, f_H=25 kHz, L=45 mH = 0.045 H

Bandwidth (BW) = $f_H - f_L = (25-23) kHz = 2 kHz$
Resonant frequency

$\begin{align*} \begin{split} & f_r = f_H - \frac{BW}{2} \\ & = 25*10^3 - \frac{2*10^3}{2} \\ & = 25000-1000 \\ & = 24*10^3 \\ & f_r = 24 kHz \end{split} \end{align*}$

At resonance,

$\begin{align*} \begin{split} & f_r = \frac{1}{2\pi \sqrt{LC}} \\ & 24*10^3 = \frac{1}{2*\pi*\sqrt{0.045*C}} \\ & C = 977 pF \end{split} \end{align*}$

Quality factor (Q) = $\frac{f_r}{BW} = \frac{24000}{2000} = 12$
Now,

$\begin{align*} \begin{split} & Q = \frac{\omega_r L}{R} \\ & R = \frac{\omega_r L}{Q} = \frac{2*\pi*f*L}{Q} \\ & = \frac {2*\pi*24000*0.045}{12} \\ & R = 565.2 \Omega \end{split} \end{align*}$

The stored design values are

R=565.2 ohms , L=45 mH , C=977 pF. Those figures match 2π√(LC) ≈ 24 kHz and R = 2π fr L / Q to the stored rounding.

The corresponding RLC notch is shown below.

Notch Filter Design Example — Design of a RLC Notch Filter

Notch Filter Transfer Function

A stored second-order notch transfer function is

$\begin{align*} H(s) = \frac{(S^2+\omega_z^2)}{S^2+\frac{\omega_p}{Q}S+\omega_p^2} \end{align*}$

where $\omega_z$ is the zero (notch) angular frequency

$\omega_p$ is the pole angular frequency

Q is the selectivity. $Q = \frac{f_r}{f_H-f_L} = \frac{f_r}{BW}$

where BW is the 3 dB stop bandwidth.

The same H(s) gives a standard notch, a low-pass notch or a high-pass notch. Which one you get depends on $\omega_z$ versus $\omega_p$ .

If $\omega_p=\omega_z$ , the filter is a standard notch.
If $\omega_p$ < $\omega_z$ , the filter is a low-pass notch (more high-frequency gain).
If $\omega_p$ > $\omega_z$ , the filter is a high-pass notch (more low-frequency gain).

Notch Filter Applications

Where notch filters are used:

Communications, instruments and biomedical front-ends use notches to cut 50 Hz or 60 Hz mains.
Electronics and radio use a band-stop to reject one band and pass the rest with little extra loss.
Switching AC & DC motor drives, converters and inverters inject tones at harmonics of the line. A notch can remove those tones so a measurement is not swamped.
In image and signal processing a notch can cut a single interfering tone.
In audio it is used to cut hum or a whistle.
Telephone and DSL plant use stop filters on known interferers. Note that DSL is the Digital Subscriber Line that is used to transmit digital information over telephone lines.
Guitar, bass and PA amps use a notch to cut a plug-in hum or a room whistle. Note that PA (Public Address) systems are an electronic system comprising of microphones, amplifiers, loudspeakers, and other musical equipment.
In ECG work a 50 Hz or 60 Hz notch cuts mains. It does not remove the DC baseline. A separate high-pass does that.