SBOA571 Application note | TI.com

SBOA571 august 2023 OPA2387 , OPA387 , OPA4387

4.1 Definition of Boundary Conditions

The pole zero plot contains fist order poles and zeros as well as two complex second order poles and zeros.

Figure 4-1 Pole-Zero-Plot of the Second Order Shelving Filter

For simplification we define:

Equation 9.

Q_{0 x 12} = Q_{p x 34} = \frac{R_{1}}{(R_{2} + 2 R_{1})} \sqrt{\frac{C_{1}}{C_{2}}} = \frac{R_{3}}{(R_{4} + 2 R_{3})} \sqrt{\frac{C_{3}}{C_{4}}}

Equation 10.

ε_{p 1} = ε_{0 x 12} = - \frac{2}{{C_{1} R}_{1}} = - \frac{(C_{2} R_{2} + {2 C}_{2} R_{1})}{2 {C_{2} C}_{1} R_{1} R_{1}} R_{2} = {2 R}_{1}

Equation 11.

ε_{p 2} = ε_{04} = - \frac{2}{{C_{2} R}_{2}} = - \frac{1}{{C_{4} R}_{4}}

Equation 12.

{D C}_{g a i n} = - \frac{R_{3}}{R_{1}}

Equation 13.

{A C}_{g a i n} = - \frac{R_{4}}{R_{2}} = - \frac{C_{2}}{C_{4}}