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)?