problem 4 outcome 2 1 find the network function its poles and zeros fo

Question

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
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e(t)
R
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(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)?