4.1

For the circuit shown in Fig. 4-1:

1. Derive Eq. (4-1).

2. If K ₂ = 2 and K ₁ = 1, Y ₁ = Y _A = 1 + s, and

   
   

   determine the transfer function H( s) and sketch the magnitude response.

3. Compare part (b) with the case where K ₁ changes by +10% and K ₂ does not change.

4. Repeat part (c) if K ₂ changes by +10% and K ₁ does not vary.

4.2

For the circuit in Fig. 4-5:

1. Derive Eq. (4-5), where K = 1 + R _a/R _b, for a noninverting op amp.

2. Design the circuit for a Butterworth response with ? _3dB = 2 ?(10 ³) rad/sec. Assume that C ₁ = C ₂ = 0.1 ?F and R _a = R _b = 1 k ?.

3. Determine the passive sensitivities of ? ₀ and Q, that is, and , where x _i x _i are the R's and C's.

4. Determine the active sensitivities and .

4.3

Determine the pole locations corresponding to the design in Prob. 4.2 if the op amp actual gain is given by Eq. (4-15) with GB = 2 ? 10 ⁵ rad/sec.

4.4

1. Repeat the design in Prob. 4.2(b) but instead of equal capacitor values, assume equal resistor values (i.e., R ₁