Band Stop Filter

Band Stop Filter:

Band Stop Filter stop a range of frequencies between two cut-off frequencies f₁ and f₂ while pass all the frequencies below f₁ and above f₂. Thus range of frequencies between f₁ and f₂ constitutes a stop band in which attenuation to the frequencies is infinite ideally. The frequencies below f₁ and above f₂ constitute two separate pass bands in which attenuation to the frequencies is zero ideally.

The Band Stop Filter can be obtained by connecting low pass filter and high pass filter sections in parallel where cut-off frequency of the low pass filter section is less than that of the high pass filter section. But the economial form of the band elimination filter can be obtained by combining the low pass and high pass filter section if series, arm contains parallel resonant circuit while shunt arm contains series resonant circuit as shown in the Fig. 9.20 (a) and (b).

The band elimination characteristics can be obtained by using conventional Band Stop Filter (either T or π type) as shown in the Fig. 9.20, if the series resonant frequency of the shunt arm is selected same as the parallel resonant frequency of the series arm. Consider ‘T’ type band elimination filter section as shown in the Fig. 9.20(a).

Let the frequency of the series and shunt arm be ω₀ rad/sec. Then for series arm, frequency of anti resonance is given by,

Similarly, for shunt arm, frequency of resonance is given by,

From equations (1) and (2), for same resonant frequencies of series and shunt arm resonant circuit we can write,

Design Impedance (R₀):

Total series arm impedance is given by,

Similarly, total shunt arm impedance Z₂ is given by,

Hence Z₁Z₂ = L₂/C₁ = L₂/C₂ which is real and constant. Hence above sections are constant k sections. So we can write,

Reactance Curves and Expressions for Cut-off Frequencies:

To verify the band elimination characteristics, let Z₁ =j X₁ and Z₂ =j X₂. The reactance curves of X₁ and X₁/4 + X₂against frequency are as shown in the Fig. 9.21.

From the above characteristics it is clear that the reactance curves for X₁ and (X₁/4 + X₂) are on the same sides of the frequency axis between f₁ and f₂ which indicates stop band. These curves are on opposite sides of the axis below f₁ and above f₂ which indicates two pass band. Hence for the given section, the characteristics are of band elimination filter where f₁ and f₂ are the cut-off frequencies.

In Band Stop Filter, the condition for cut-off frequencies is given by

But from the condition of constant K filter section, Z₁Z₂ = R²₀

From above equation it is clear that the value of the series arm impedance Z₁ can be obtained at two different cut-off frequencies namely f₁ and f₂. So at f = f₁, Z₁ = + j(2 R₀) and at f = f₂, Z₁= -j(2 R₀). Thus impedance Z₁ at f₁, i.e. at lower cut-off frequency, is negative of the impedance Z₁ at f₂ i.e. upper cut-off frequency. Hence we can write,

But from equation (1), frequency of resonance is given by

Substituting value of (L₁ C₁ ) in above equation, we can write,

Simplifying above equation,

Hence, above equation (9) indicates that in band elimination filter, the frequency of resonance of the individual arms is the geometric mean of two cut-off frequencies.

Variation of Z_0T and Z_0π, Attenuation Constant (α), Phase Constant (β) with Frequency:

The variations of Z_0T and Z_0π, attenuation constant (α) and phase shift (β) are as shown in the Fig. 9.22 (a), (b) and (c) respectively. Consider that f₁ and f₂ are two cut-off frequencies and R₀ is the design impedance of the Band Stop Filter.

Design Equations:

Consider that a band elimination filter with two cut-off frequencies f₁ and f₂ is terminated in design impedance R₀. Then, from equation (7), at lower cut-off frequency f₁, we can write,