Question 9.2: Objective: An op-amp with a T-network is to be designed as a...

We need a voltage gain of
|A_{v}| = \frac{1.2}{0.012} = 100
Equation (9.18) can be written in the form

A_{v} = \frac{v_{O}}{v_{I}} = − \frac{R_{2}}{R_{1}} \left(1 + \frac{R_{3}}{R_{4}} + \frac{R_{3}}{R_{2}} \right)         (9.18)
A_{v} = − \frac{R_{2}}{R_{1}} \left(1 + \frac{R_{3}}{R_{4}} \right)  −  \frac{R_{3}}{R_{1}}
If, for example, we arbitrarily choose \frac{R_{2}}{R_{1}} = \frac{R_{3}}{R_{1}} = 8 , then
−100 = −8 \left( 1 + \frac{R_{3}}{R_{4}} \right)  −  8

which yields
\frac{R_{3}}{R_{4}} = 10.5
The effective R_{1} must include the R_{S} resistance of the microphone. If we set R_{1} = 49  k\Omega so that R_{1,eff} = 50  k\Omega, then
R_{2} = R_{3} = 400  k \Omega,
and
R_{4} = 38.1  k \Omega,
Design Pointer: If we need to use standard resistance values in our design, then, using Appendix C, we can choose R_{1} = 51  k\Omega, so that R_{1,eff} = 52  k\Omega, and we can choose R_{2} = R_{3} = 390  k\Omega. Then, using Equation (9.18), we have
A_{v} = −100 = \frac{−R_{2}}{R_{1,eff}} \left(1 + \frac{R_{3}}{R_{4}} \right)  −  \frac{R_{3}}{R_{1,eff}} = \frac{−390}{52} \left(1 + \frac{390}{R_{4}} \right)  −  \frac{390}{52}
which yields R_{4} = 34.4  k \Omega. We may use a standard resistor of R_{4} = 33  k \Omega. This resistance value then produces a voltage gain of A_{v} = −103.6 .
Trade-offs: If we consider ±2 percent tolerances in the standard resistor values, the voltage gain can be written as
A_{v} = \frac{−R_{2}(1  ±  0.02)}{1  k \Omega  +  R_{1}(1 ± 0.02)} \left[ 1  +  \frac{R_{3}(1  ±  0.02)}{R_{4}(1  ±  0.02)} \right]  −  \frac{R_{3}(1  ±  0.02)}{1  k \Omega  +  R_{1}(1  ±  0.02)}
or

A_{v} = \frac{−390(1  ±  0.02)}{1  +  51(1  ±  0.02)} \left[ 1  +  \frac{390(1  ±  0.02)}{33(1  ±  0.02)} \right]  −  \frac{390(1  ±  0.02)}{1  +  51(1  ±  0.02)}
Analyzing this equation, we find the maximum magnitude as |A_{v}|_{max} = 111.6 or +7.72 percent, and the minimum magnitude as |A_{v}|_{min} = 96.3 or −7.05 percent.
Comment: As required, all resistor values are less than 500 kΩ. Also the resistance ratios in the voltage gain expression are approximately equal. As with most design problems, there is no unique solution. We must keep in mind that, because of resistor value tolerances, the actual gain of the amplifier will have a range of values