(a) Perform a Monte Carlo analysis of the six-stage cascade amplifier design resulting from the exam- ple in Tables 12.4 and 12.5, and determine the frac- tion of the amplifiers that will not meet either the gain or bandwidth specifications. Assume the resis- tors are uniformly distributed between their limits. A, > 100 dB fu 2 50 kHz and (b) What tolerance must be used to ensure that less than 0.1 percent of the amplifiers fail to meet both specifications? The equation here can be used to estimate the location of the half-power frequency for N closely spaced poles, where fHi is the average of the indi- vidual cutoff frequencies of the N stages and f is the cutoff frequency of the ith individual stage. fH = FHIV2/N – 1 where THi =E fin n TABLE 12.5 Cascade of Six Identical Noninverting Amplifiers Av (0) GAIN PER fH1 SINGLE STAGE NUMBER N STAGE OF STAGES STAGE 1/B fH N STAGES GAIN B x fr Rin Rout 6.81E + 00 1.00E + 05 6.83E + 00 1.02E + 05 6.85E + 00 1.04E + 05 1.47E + 05 1.46E + 05 5.121E + 04 1.46E + 13 3.42E - 03 1.46E + 05 5.137E + 04 1.47E + 13 3.41E – 03 5.107E + 04 1.46E + 13 3.43E – 03 6.87E + 00 1.05E + 05 6.89E + 00 1.07E + 05 6.91E + 00 1.09E + 05 6.93E + 00 1.11E + 05 6.95E + 00 1.13E + 05 1.45E + 05 5.092E + 04 1.46E+ 13 3.44E – 03 1.45E + 05 6. 5.077E + 04 1.45E + 13 3.45E – 03 5.062E + 04 1.45E + 13 3.46E – 03 6. - 1.45E + 05 1.44E + 05 5.048E + 04 1.44E+ 13 3.47E –- 03 1.44E + 05 6. - 5.033E + 04 1.44E + 13 3.48E – 03 6.97E + 00 1.15E + 05 1.43E + 05 5.019E + 04 1.43E + 13 3.49E – 03 5.004E + 04 1.43E + 13 3.50E – 03 6.99E + 00 1.17E + 05 7.01E + 00 1.19E + 05 1.43E + 05 1.43E + 05 4.990E + 04 1.43E + 13 3.51E – 03 TABLE 12.4 Design of Cascade of N Identical Operational Amplifier Stages CASCADE OF IDENTICAL NONINVERTING AMPLIFIERS Ay (0) GAIN PER STAGE fн1 SINGLE STAGE NUMBER fy N STAGES OF STAGES 1/8 B x fr Rin Rout 1.00E + 05 3.16E + 02 4.64E + 01 1 1.00E + 01 1.000E + 01 2.00E + 09 2.50E + 01 3.16E + 03 2.15E + 04 2.035E + 03 1.098E + 04 3.17E+ 11 2.16E + 12 1.58E – 01 3 2.32E – 02 5.62E + 12 1.78E + 01 1.00E + 01 4 5.62E + 04 2.446E + 04 8.89E – 03 1.00E + 05 3.856E + 04 5.137E + 04 6.229E + 04 7.134E + 04 1.00E + 13 1.47E + 13 1.93E + 13 5.00E – 03 - 1.47E + 05 1.93E + 05 6. 6.81E + 00 5.18E + 00 4.22E + 00 3.41E – 03 2.59E – 03 8. 2.37E + 05 2.37E + 13 2.11E – 03 CASCADE OF IDENTICAL INVERTING AMPLIFIERS NUMBER OF STAGES Ay(0) fн1 SINGLE STAGE fH N STAGES (1/B) – 1 Rin Rout 1.00E + 05 3.16E + 02 4.64E + 01 R1 2.50E + 01 1.00E + 01 3.15E + 03 2.11E + 04 1.00E + 01 2.03E + 03 1 1.58E – 01 R1 R1 1.08E + 04 2.32E – 02 R1 1.78E + 01 1.00E + 01 6.81E + 00 5.32E + 04 2.32E + 04 3.51E + 04 4.48E + 04 4 8.89E – 03 - 9.09E + 04 1.28E + 05 R1 R1 5.00E – 03 6. 3.41E – 03 5.18E + 00 4.22E + 00 1.62E + 05 5.22E + 04 5.77E + 04 R1 R1 2.59E – 03 1.92E + 05 2.11E – 03