Band-Stop Filters

A Band-Stop Filter is a circuit that allows most frequencies to pass, but blocks or attenuates a certain range or band of frequencies. It is also known as a 'band-elimination' filter or a 'band-rejection filter'. The band-stop filter is the opposite of the band-pass filter.

The range of frequencies that a band-stop filter blocks is known as the 'stopband', which is bound by a lower cut-off frequency f1 and a higher cut-off frequency f2. A special type of band-stop filter, known as the 'notch filter', is one whose stopband is very narrow, thus creating a 'notch' in the frequencies allowed to pass. The notch filter is therefore a band-stop filter that has a high Q factor. Combining several notch filters together forms a 'comb filter', which is a filter that has multiple stopbands.

An ideal band-stop filter is one whose stopband is completely rejected by it, while allowing all other frequencies to pass unchanged (no gain nor attenuation). In an ideal band-stop filter, the transition of the response from outside the stopband to within the stopband and vice versa is instantaneous. Of course, an ideal notch filter doesn't exist in the real world, i.e., complete attenuation within the stopband can not be achieved while frequencies outside the stopband undergo some level of attenuation.

The over-all impedance of a resonant series LC circuit consisting of an inductor and a capacitor in series with each other will drop to zero at the resonant frequency because the reactances of the inductor and the capacitor cancel each other out under resonance. On the other hand, the over-all impedance of a resonant parallel LC circuit consisting of an inductor and a capacitor in parallel with each other will increase to infinity at the resonant frequency, i.e., the reactances of the inductor and the capacitor result in zero current flow under resonance.

Resonant series and parallel LC circuits may thus be combined to form a notch filter as shown in Figure 1. In this circuit, the resonant parallel LC circuits are used to block frequencies within the stopband, while the resonant series LC circuit is used to attenuate frequencies within the stopband by shunting them towards the ground.

Figure 1. A Notch Filter Using Resonant LC Circuits

The following equations apply to the notch filter in Figure 1 above:

1) L = Zo (f2-f1) / (πf1f2); Lf = Zo / (4π(f2-f1))

2) C = 1 / (4π(f2-f1)Zo); Cf = (f2-f1) / (π(f2f1Zo))

3) fo = sqrt(f1f2) = 1 / (2π(sqrt(LC))) = 1 / (2π(sqrt(LfCf)))

4) Zo = sqrt(L/Cf) = sqrt(Lf/C)

where Zo = line impedance; f1 = lower cut-off frequency; f2 = upper cut-off frequency; fo = resonant frequency.