Problem 4 Outcome 2.1: Find the network function, its poles and zeros, for a given linear circuit. Outcome 2.4: Select element values to produce specific poles and zeros of a given linear circuit. R . www e(t) R www (c) Use V₂(s) V+(s) to show that Q Vi[s) V₁(8) RA The circuit above, the Sallen-Key circuit, will be integral to our filter design practices later in the semester. (a) Label the voltage at the non-inverting terminal V+(s). Argue that the output voltage is V₂(8) = µV+(8) and find in terms of RA and Rn. Hint: Use ideal op-amp assumptions. Ra (b) Label the node between the two resistors R as Vx(s). Write KCL both at that node and at the non- inverting input, V. (s), and solve your KCL equations for V+(s) to show that (1/RC)² ¡V₁(s) s²+ s(3-µ)(1/RC) + (1/RC)² P/(RC)² = 8² + x(3 − µ)(1/RC) + (1/RC)² (d) With R = R₁ = 100 kft, choose R and C such that the circuit implements 2 x 105 H(s) = ²+1 × 10³s +1 × 10⁰ (e) Where are the poles and zeros of the transfer function of part (d)?

Fig: 1