Question 15.DA.7: AN ACTIVE BANDPASS FILTER Objective: • Design an active band...

(Analysis): Considering the circuit in Figure 15.53, we have
\frac{v_{o2}}{v_{o}} = − \frac{\frac{1}{s C}}{R_{2}} = \frac{−1}{s R_{2}C}

and
\frac{v_{o3}}{v_{o2}} = −1
so
v_{o3} = \frac{v_{o}}{s R_{2}C}                (15.128)
Node 1 is at virtual ground. Summing currents at this node, we find
\frac{v_{i}}{R_{4}} + \frac{v_{o}}{R_{1}} + \frac{v_{o}}{\frac{1}{s C}} + \frac{v_{o3}}{R_{3}} = 0
Substituting the expression for v_{o3} from Equation (15.128), we have
\frac{v_{i}}{R_{4}} = −v_{o} \left(\frac{1}{R_{1}} + s C + \frac{1}{s R_{2} R_{3} C} \right)

The overall voltage gain is
\frac{v_{o}}{v_{i}} = \frac{\frac{−1}{R_{4}}}{\left(\frac{1}{R_{1}} + sC + \frac{1}{s R_{2} R_{3} C} \right)}

Setting s = jω to obtain the steady-state frequency response, we obtain
\frac{v_{o}}{v_{i}} = \frac{\frac{−1}{R_{4}}}{\left[\frac{1}{R_{1}} +  j \left(ωC  −  \frac{1}{ω R_{2} R_{3} C} \right) \right] }

The center frequency occurs at the point where the imaginary term in the denominatoris zero, or
ω_{o} C = \frac{1}{ω_{o} R_{2} R_{3} C}
which can be rewritten as
f_{o} = \frac{1}{2π C \sqrt{R_{2}R_{3}}}

The maximum voltage gain occurs at the center frequency, so that
|A_{v}|_{max} = \frac{R_{1}}{R_{4}}
The bandwidth is given by
BW = \frac{1}{2π R_{1} C}
(Design): If we let C = 0.1 µF, then we can find
R_{1} = \frac{1}{2π (BW)C} = \frac{1}{2π (10) (0.1  ×  10^{−6})} = 159  k \Omega

From the maximum gain, we determine
|A_{v}|_{max} = \frac{R_{1}}{R_{4}} ⇒ 40 = \frac{159}{R_{4}}
or
R_{4} = 3.975  k \Omega
If we choose R_{2} = R_{3}, then from the center frequency
f_{o} = \frac{1}{2π C \sqrt{R_{2} R_{3}}}
we find
R_{2} = R_{3} = \frac{1}{2π f_{o} C} = \frac{1}{2π(2  ×  10^{3})(0.1  ×  10^{−6})}
or
R_{2} = R_{3} = 795.8  \Omega
(Standard Resistor Values): The closest standard resistor values are R_{2} = 750 Ω, R_{3} = 820 Ω, R_{1} = 160  kΩ, and R_{4} = 3.9  kΩ. A capacitor of 0.1 µF is a standard value. Using these circuit elements, we find the center frequency to be
f_{o} = \frac{1}{2π C \sqrt{R_{2} R_{3}}} = \frac{1}{2π(0.1  ×  10^{−6}) \sqrt{(750)(820)}}
or
f_{o} = 2.029  kHz
The bandwidth is
BW = \frac{1}{2π R_{1}C} = \frac{1}{2π(160  ×  10^{3})(0.1  ×  10^{−6})}
or
BW = 9.947 Hz

The maximum voltage gain at the center frequency is
|A_{v}|_{max} = \frac{R_{1}}{R_{4}} = \frac{160}{3.9} = 41.03

Comment: Using standard resistor values, the center frequency is within 1.5 percent of the design specification, the bandwidth is within 0.53 percent of the design specification, and the maximum gain is within 2.6 percent of the design specification. The circuit elements, of course, have tolerances that will affect the final circuit performance.