a) Show that the transfer function for a prototype narrow band bandreject filter isb) Use...

a) Show that the transfer function for a prototype narrow band bandreject filter is

b) Use the result found in (a) to find the transfer -function of the filter designed in Example 15.13.Example 1

Designing a High-Q Bandreject Filter

Design a high-Q active bandreject filter (based on the circuit in Fig. 15.29) with a center frequency of 5000 rad/s and a bandwidth of 1000 rad/s. Use 1 µF capacitors in your design.

Solution

In the bandreject prototype filter, ωO= 1 rad/s, R = 1 Ω, and C = 1 F. As just discussed, once ωO and Q are given, C can be chosen arbitrarily, and R and a can be found from Eqs. 1 and 2. From the specifications, Q = 5. Using Eqs. 15.68 and 15.69, we see that

R = 200 Ω,

σ = 0.95.

Therefore we need resistors with the values 200 Ω (R), 100 Ω (R /2), 190 Ω (σR),and 10 Ω [(1 - σ)R]. The final design is depicted in Fig. 1, and the Bode magnitude plot is shown in Fig. 2.

Figure 1

Figure 2

Equation 1

Equation 2