Notch Filter (Band-Stop): What is it? (Circuit, Design & Transfer Function)

What Is A Notch Filter Band Stop

What is a Notch Filter (Bandstop Filter)?

A notch filter (also known as a bandstop filter or reject filter) is defined as a device that rejects or blocks the transmission of frequencies within a specific frequency range and allows frequencies outside that range. Notch filters eliminate transmission of a narrow band of frequencies and allow transmission of all the frequencies above and below this band. As it eliminates frequencies hence, it is also called a band elimination filter.

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

If the filter is wideband, it is referred to as a band-reject filter and if the filter is narrow-band, it is referred to as a notch filter. The characteristics of a band-stop filter are exactly the reverse of the bandpass filter. Hence, the notch filter is a complement of the bandpass filter.

For example, if a Notch Filter has a stopband frequency from 100 MHz to 200 MHz, then it will pass all the signals from DC to frequency of 100 MHz and above 200 MHz, it will only reject frequency between 100 MHz to 200 MHz.

Thus, the function of a Notch Filter is to passing all those frequencies from zero (DC) up to lower cut-off frequency(fL) and above higher cut-off frequency(fH), and reject all those frequencies that lie in the bandwidth region i.e., BW= fH-fL.

Detection and filtering of narrowband signals in the presence of noise are the important applications of signal processing techniques. In many applications, it is necessary to remove the narrowband signal without changing band energy. This can be achieved by passing the signals through a notch filter.

Notch Filter Circuit

The notch filter is a combination of low pass and high pass filters similar to the bandpass filter design, but the difference is that we connect both the filters in parallel connection instead of cascading connection. The circuit diagram of the notch filter is shown in the below figure.

Passive Notch Filter Circuit
Notch Filter Circuit

The upper portion of the notch filter circuit is a passive RC low pass filter. This portion comprises two resistors R1, R2,and capacitor C1 in the form of a ‘T’ configuration. This filter will allow the signals having frequencies lower than the higher cut-off frequency (fH).

The lower portion of the notch filter circuit is a passive RC high-pass filter. This portion comprises two capacitors C2, C3,and resistor R3 also in the form of ‘T’ configuration. This filter will allow the signals having frequencies higher than the lower cut-off frequency (fL). This combination of both ‘T’ configuration is commonly known as a ‘Twin-T’ filter.

The typical configuration and a frequency response of a notch filter is shown in the figure below.

Typical Notch Filter Configuration
Typical Configuration of a Notch Filter
Frequency Response of A Notch Filter
Frequency Response

Notch Filter Types

There are many types of notch filter circuits are designed. Let’s explain the major types of filter circuits in detail.

Active Notch Filter

The active notch filter is a parallel combination of low pass filter and high pass filter with the op-amp as an amplifying component is shown in the figure below.

Active Notch Filter Circuit
Active Notch Filter Circuit

The circuit diagram of the Active Notch Filter is divided into three portions. The upper portion is an active low pass filter (LPF) which is in parallel to an active high-pass filter (HPF). The third and last portion of the circuit is the amplifying part in which op-amp is used for amplification. The final op-amp stage is a summing amplifier or a Op-Amp adder which acts as a summer. The circuit diagram of an active notch filter is shown in the figure below.

Here, the high-frequency cut-off of the low-pass filter is

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

while the low-frequency cut-off of the high-pass filter is

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

Passive Notch Filter

The passive filter comprises only passive components such as resistors, inductors, and capacitors and it does not use any active components such as op-amp for amplification. So, the amplification part is absent in a passive notch filter.  

The passive notch filter is a combination of a passive low pass filter and a passive high pass filter. The circuit diagram of the passive notch filter is shown in the figure below.

Passive Notch Filter Circuit
Passive RC Notch Filter Circuit

The upper portion of the passive notch filter circuit is the passive low pass filter in parallel with the passive high pass filter.

Optical Notch Filter

A Notch Filter is optical filters that selectively reject a portion of the spectrum while transmitting all other wavelengths. It can be designed for various rejection levels, which is typically in terms of optical density.

In an optical system, specific wavelengths of light are selectively redirected by a diffraction grating or a dispersive prism. In the case of transmission gratings and prisms, polychromatic light that passes through the object will be redirected to wavelength, and then desired wavelengths can be achieved by the optical notch filter.

In the case, when using optics with real materials, the light will be attenuated at various wavelengths and it is filtered by interference with the medium through which the light passed. An optical notch filter passes wavelengths of a light that are unaltered or minimally attenuated.

Optical notch filters are used in laser-based Raman spectroscopy, laser-based fluorescence, and other biomedical and life science applications.

RLC Notch Filter

As the name suggests RLC, this notch filter or bandstop filter contains passive elements resistor, inductor, and capacitor. Therefore, it is also known as a passive notch filter. The circuit diagram of the RLC notch filter is shown in the below figure.

RLC Notch Filter Circuit
RLC Notch Filter Circuit

The shunt element of an RLC notch filter is a series combination of L and C. The output is taken across this series combination of inductor and capacitor. The equivalent circuit and characteristics of the RLC notch filter are shown in the figure below.

RLC Notch Filter Equivalent Circuit
RLC Notch Filter Equivalent Circuit

Here, the equivalent impedance of a series combination of the L and C is given by,

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

Now, the net impedance of the circuit is given by,

    \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*}

Now, applying KVL (Kirchhoff’s Voltage Law) in loop a-b-c-d to the equivalent circuit of RLC notch filter, we get,

    \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*}

Now, at the series resonance condition, the resonant frequency is given by

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

And the Q-factor is given by,

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

Where, \omega_r = Bandwidth = \frac{f_r}{Q}

while, f_L = (f_r - \frac{BW}{2}) Hz is the lowest cut-off frequency, and

f_H = (f_r + \frac{BW}{2}) Hz is the highest cut-off frequency. The characteristics of the notch filter is shown in the figure below.

Notch Filter Characteristics
Characteristics of a Notch Filter

RLC type Notch Filter can also be made by utilizing the circuit that is shown in the figure below.

RLC Notch Filter Circuit 2
RLC Notch Filter Circuit

RF Notch Filter 

A narrow notch filter is an RF notch filter that can be used to reject the carrier frequency during the measurement of non-linearities of power amplifiers.

An LC circuit notch filter is used to reject a specific interfering frequency in the radio frequency domain. The LC circuit notch filter circuit is used with radio receivers that are close to the transmitter that it swamps all other signals. The wave trap is used to reject the signal from the transmitter.

FM broadcast signals are very strong and it can prevent an SDR (Software-defined Radio) from processing weak signals. FM notch filters are very useful for SDR applications. Hence, The FM signals first pass through the notch filter, and then it is given to the SDR USB.

RF Notch Filter Configuration
RF Notch Filter Configuration

Butterworth Notch Filter

A Butterworth notch filter is a special type of signal processing filter designed to have a frequency response as flat as possible as in the narrow stopband.

An ideal Notch filter rejects some range of frequencies without distortion and passes all other frequencies. But in some applications, a simple notch filter is not accurate and reliable because of the low signal to noise ratio. In those cases, Butterworth’s notch filter is used to increase accuracy and reliability.

For example, In ECG (electrocardiogram) the noises that commonly disturb are power line interference, instrumentation noise, external magnetic field interference, noise due to random body movements, and respiration movements. These noises can be characterized based on their frequency content.

The bandwidth of the noise overlaps that of the desired signals so that ordinary filtering technique cannot sufficiently enhance the signal to noise ratio. Therefore, it is necessary to reduce these noises in the ECG signal to improve accuracy and reliability. The 4th order Butterworth notch filters with 3 dB stopband are used to reduce 50 Hz power line noise interference from ECG signals.

Notch Filter Design Example

An RLC type Notch Filter is to work with cut-off frequencies 23 kHz and 25 kHz. Assume Inductance L=45 mH, Design the RLC type Notch Filter.

Given data: fL=23 kHz, fH=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*}

  • Now, at resonance condition resonant frequency is given by,

    \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*}

Hence, the Notch filter configurations are

R=565.2 ohms , L=45 mH , C=977 pF

Thus, the design of the notch filter from the above configurations is shown in the below figure.

Notch Filter Design Example
Design of a RLC Notch Filter

Notch Filter Transfer Function

The transfer function of the Notch Filter is given by

    \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-circular frequency

\omega_p is the pole-circular frequency

Q is the Quality factor. It is the selectivity of the filter. Q = \frac{f_r}{f_H-f_L} = \frac{f_r}{BW}

where, BW is the bandwidth of the filter.

There are three cases in the characteristics of the notch filter. i.e., a standard notch, a low-pass notch, and a high-pass notch. The relationship between the zero-circular frequency \omega_z and the pole-circular frequency \omega_p determines the characteristics of the notch filter.

  1. If the pole-circular frequency is equal to the zero-circular frequency, i.e., \omega_p=\omega_z, then the filter is the standard notch type.
  2. If the pole-circular frequency is less than the zero-circular frequency, i.e., \omega_p < \omega_z, then the filter is the low-pass notch type.
  3. If the pole-circular frequency is greater than the zero-circular frequency, i.e., \omega_p > \omega_z, then the filter is the high-pass notch type.  

Notch Filter Applications

Some of the applications of the Notch Filter include:

  • A Notch filter is generally used in communication systems, Instrumentation and control systems, and the biomedical field to eliminate 50/60 Hz power line interference.
  • Notch filter or Bandstop filter is widely used in electronics and communications circuits to reject a band of unwanted frequencies and allowing transmission of other frequencies with minimum loss.
  • Switching type of AC & DC motor drives, converters, and inverters cause sinusoidal disturbances at certain harmonics of the line frequency. The use of a notch filter eliminates such unwanted disturbances and enables accurate measurements.
  • It is highly preferred in image and signal processing to reject unwanted frequencies i.e. noise.
  • It is used in audio signal processing, for removing a specific range of unwanted frequencies i.e. noise or hum.
  • It is used in telephone technology, DSL, and other internet services as a line noise reducer to reduce unwanted interference. Note that DSL is the Digital Subscriber Line that is used to transmit digital information over telephone lines.
  • It is used in guitar amplifiers, instrument amplifiers, acoustic guitar, mandolin, bass instrument amplifier, and PA systems to reduce a specific humming sound that may produce after instruments are plugged. Note that PA (Public Address) systems are an electronic system comprising of microphones, amplifiers, loudspeakers, and other musical equipment.
  • It is used in medical field applications i.e. in ECG (Electrocardiogram) measurements, to eliminate dc component.
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