Bandstop filters and Notch filters

What is a bandstop or bandreject filter?

A bandstop filter, or a band-reject filter, is an electronic or signal processing filter designed to attenuate a specific range of frequencies within a signal while allowing all other frequencies to pass through relatively unaltered. It is opposite to a bandpass filter, where the bandpass filter passes a specific range of frequencies. In other words, it rejects or attenuates a particular band of frequencies while passing frequencies outside that band, as shown in Fig 1. A very sharp bandstop filter is sometimes called a notch filter, which removes a very narrowband of frequency. 

Frequency response of Bandstop filter

bandstop filter transfer function plot
Fig 1 : Gain and Phase plot of a bandstop/bandreject filter.
The frequency response of a bandstop filter (Fig 1) shows how the filter attenuates or rejects certain frequencies while allowing others to pass through. A bandstop filter is designed to attenuate a specific range of frequencies while allowing frequencies outside that range to pass relatively unaltered. The frequency response of a bandstop filter typically has the following characteristics:
  1. Passband: Frequencies outside the specified range (the stopband) are allowed to pass with minimal attenuation. In this region, the filter has a flat or nearly flat response. These frequencies are considered the “passband” of the filter.
  2. Stopband: Frequencies within the specified range to be attenuated are heavily attenuated. The degree of attenuation within the stopband depends on the design and specifications of the filter. The stopband is the range of frequencies that the filter is designed to block or attenuate.
  3. Center Frequency (fC): The center frequency of the notch or bandstop is the frequency at which the filter provides maximum attenuation. This is the frequency that the filter is primarily designed to reject.
  4. Bandwidth: The bandwidth of the bandstop filter is the range of frequencies over which attenuation occurs. It is typically defined as the width of the stopband at -3 dB of attenuation.
  5. Roll-off rate: The roll-off is the rate at which the filter’s gain decreases as we move away from the center frequency (fc) and into the stopband. The roll-off rate is usually specified in decibels per decade (dB/decade). A steeper roll-off indicates a faster attenuation rate.
  6. Q Factor (Quality Factor): The Q factor is a measure of the sharpness or selectivity of the bandstop filter. A higher Q factor results in a narrower bandwidth and a sharper notch at the center frequency. Conversely, a lower Q factor results in a wider bandwidth and a shallower notch.

Simple active bandstop filter

bandstop filter using opamp
Fig 2 : An active bandstop / bandreject filter.
Fig 2 shows an active bandreject filter. It consists of a high-pass filter and a low-pass filter in parallel. The cutoff frequency of the low pass filter is fL. The cutoff frequency of the high pass filter is fH. To make a band-reject / band-stop filter, fL < fH as shown in frequency response in Fig 1. Since it is an active filter, a gain can also be obtained in the pass-band.

What is a notch filter?

A notch filter is a special band-stop filter that attenuates or suppresses a specific frequency while allowing other frequencies to pass without significant attenuation. It is designed to create a “notch” or a deep attenuation in the frequency response at a particular frequency, as shown below.

notch filter circuit's gain vs frequency response and transfer function
Fig 3 : Frequency response (Gain and Phase) of a notch filter.

Passive notch filter

A passive notch filter attenuates a narrow range of frequencies while allowing all other frequencies to pass through relatively unaltered without requiring an external power source or active components like amplifiers or transistors. Instead, it relies on passive components such as resistors, capacitors, and inductors to achieve the desired frequency response.

Parallel RLC notch filter

parallel passive RLC notch filter
Fig 4 : A passive parallel-RLC bandstop filter
A parallel LC passive notch filter is shown in Fig 4. At the frequency of ωc=1/√LC the LC tank’s impedance will become infinite. At this resonant frequency (fc), no current flows, and Vout is pulled to the ground by resistor R. At lower frequencies, the inductor shorts the Vout to Vin creating a passband. At higher frequencies, the capacitor shorts the Vout to Vin. The frequency response is similar to one mentioned in Fig 3.

Series RLC notch filter

series RLC passive notch filter circuit
Fig 5 : A passive series-RLC bandstop filter

A series LC passive notch filter is shown in Fig 5. At the frequency of ωc=1/√LC the LC tank’s impedance will become zero. At this resonant frequency (fc), Vout is shorted to the ground by the LC tank. At lower frequencies, the capacitor becomes open. At higher frequencies, the inductor opens. The frequency response is similar to the one mentioned in Fig 3.

Twin-T notch filter

twin T passive notch filter circuit.
Fig 6 : A passive twin-T notch filter

A twin-T notch filter is a passive filter circuit used to attenuate a specific frequency from an input signal while allowing all other frequencies to pass through. It has a simple and elegant design with a very low Bill of materials (BOM).

The name “twin-T” comes from the configuration of two “T” shaped RC (resistor-capacitor) networks connected in parallel as shown in Fig 6. The twin-T notch filter is commonly used to filter out interference or noise at a specific frequency, such as 60 Hz hum in power lines or unwanted harmonics/interference in audio signals.

The basic twin-T notch filter circuit consists of the following components:

  1. T-network 1: It is made using two resistors and a capacitor. To understand, it forms a low pass filter. The T-network makes sure that it is a bilateral network. A bilateral network is one in which the relation between current and voltage does not change in either direction in the network.
  2. T-network 2: It is made using two capacitors and a resistor. To understand, it forms a high pass filter.

This Twin-T notch filter can be used to make a oscillator as shown in the article – Twin T notch filter based oscillator

Active twin-T notch filter complete circuit

active RC notch filter circuit with adjustable Q-factor

The Q of the passive twin-T notch filter is low (<1/2). So, an active twin-T notch filter is used to obtain a very high Q. The Q can be raised from the usual 0.3 to something greater than 10. The buffer (using opamp A2) provides a low output resistance; and the high input resistance of the opamp A1 makes it possible to use large resistance values in the ‘‘T’’ network so that only small capacitors are required, even at low frequencies. The addition of opamp overcomes parasitic capacitances which makes it suitable for high-frequency operation. The frequency of the notch is unchanged with the addition of opamps. A 60dB notch can be obtained if 0.1% tolerance resistors and 1% tolerance capacitors are used.

In applications where the rejected signal might deviate slightly from the null of the notch network, it is advantageous to lower the Q of the network. This ensures some rejection over a wider range of input frequencies.

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