**Band Stop Filter:**

Band Stop Filter stop a range of frequencies between two cut-off frequencies f_{1} and f_{2} while pass all the frequencies below f_{1} and above f_{2}. Thus range of frequencies between f_{1} and f_{2} constitutes a stop band in which attenuation to the frequencies is infinite ideally. The frequencies below f_{1} and above f_{2} 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 ω_{0} 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 (R**_{0}):

_{0}):

Total series arm impedance is given by,

Similarly, total shunt arm impedance Z_{2} is given by,

Hence Z_{1}Z_{2} = L_{2}/C_{1} = L_{2}/C_{2} 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 Z_{1} =j X_{1} and Z_{2} =j X_{2}. The reactance curves of X_{1} and X_{1}/4 + X_{2 }against frequency are as shown in the Fig. 9.21.

From the above characteristics it is clear that the reactance curves for X_{1} and (X_{1}/4 + X_{2}) are on the same sides of the frequency axis between f_{1} and f_{2} which indicates stop band. These curves are on opposite sides of the axis below f_{1} and above f_{2} which indicates two pass band. Hence for the given section, the characteristics are of band elimination filter where f_{1} and f_{2} 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, Z_{1}Z_{2} = R^{2}_{0}

From above equation it is clear that the value of the series arm impedance Z_{1} can be obtained at two different cut-off frequencies namely f_{1} and f_{2}. So at f = f_{1}, Z_{1} = + j(2 R_{0}) and at f = f_{2}, Z_{1 }= -j(2 R_{0}). Thus impedance Z_{1} at f_{1}, i.e. at lower cut-off frequency, is negative of the impedance Z_{1} at f_{2} i.e. upper cut-off frequency. Hence we can write,

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

Substituting value of (L_{1} C_{1} ) 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:

_{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 f_{1} and f_{2} are two cut-off frequencies and R_{0} is the design impedance of the Band Stop Filter.

**Design Equations:**

Consider that a band elimination filter with two cut-off frequencies f_{1} and f_{2} is terminated in design impedance R_{0}. Then, from equation (7), at lower cut-off frequency f_{1}, we can write,

For band elimination filter constant K section, frequency of resonance in series arms is given by,

Substituting value of L_{1} in above equation,

From equation (6) we can write,

Substituting value of L_{1} from equation (10),

Similarly from equation (6) we can write,

Substituting value of C_{1} from equation (11),

Equations (10) to (13) are called design equations of prototype band elimination filter sections.