Chapter 8 Solutions - Soln0864 080809

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Chapter 8, Solution 64. Using Fig. 8.109, design a problem to help other students to better understand second-order op amp circuits. Although there are many ways to work this problem, this is an example based on the same kind of problem asked in the third edition. Problem Obtain the differential equation for v o (t) in the network of Fig. 8.109. Figure 8.109 Solution C2 R2 R1 vs 1 v1 C1 2  + vo At node 1, (v s – v 1 )/R 1 = C 1 d(v 1 – 0)/dt or v s = v 1 + R 1 C 1 dv 1 /dt At node 2, C 1 dv 1 /dt = (0 – v o )/R 2 + C 2 d(0 – v o )/dt or From (1) and (2), or –R 2 C 1 dv 1 /dt = v o + R 2 C 2 dv o /dt (1) (2) (v s – v 1 )/R 1 = C 1 dv 1 /dt = –(1/R 2 )(v o + R 2 C 2 dv o /dt) v 1 = v s + (R 1 /R 2 )(v o + R 2 C 2 dv o /dt) (3) Substituting (3) into (1) produces, v s = v s + (R 1 /R 2 )(v o + R 2 C 2 dv o /dt) + R 1 C 1 d{v s + (R 1 /R 2 )(v o + R 2 C 2 dv o /dt)}/dt = v s + (R 1 /R 2 )(v o )+ (R 1 C 2 )dv o /dt + R 1 C 1 dv s /dt + (R 1 R 1 C 1 /R 2 )dv o /dt + ((R 1 )2 C 1 C 2 )[d2v o /dt2] ((R 1 )2 C 1 C 2 )[d2v o /dt2] + [(R 1 C 2 ) + (R 1 R 1 C 1 /R 2 )]dv o /dt + (R 1 /R 2 )(v o ) = –R 1 C 1 dv s /dt Simplifying we get, [d2v o /dt2] + {[(R 1 C 2 ) + (R 1 R 1 C 1 /R 2 )]/((R 1 )2 C 1 C 2 )}dv o /dt + {(R 1 /R 2 )(v o )/ ((R 1 )2 C 1 C 2 )} = –{R 1 C 1 /((R 1 )2 C 1 C 2 )}dv s /dt d2v o /dt2 + [(1/ R 1 C 1 ) + (1/(R 2 C 2 ))]dv o /dt + [1/(R 1 R 2 C 1 C 2 )](v o ) = –[1/(R 1 C 2 )]dv s /dt Another way to successfully work this problem is to give actual values of the resistors and capacitors and determine the actual differential equation. Alternatively, one could give a differential equations and ask the other students to choose actual value of the differential equation.