Solve Quadratic Equations for Analog Design Engineers

the story of a new formula

Almost everyone learns in school, how to solve a quadratic equation using this formula

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  • use a single quantity, such as “q-factor” to depict a wide range of behavior of a complicated circuitry
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  • we use F=1, due to Q much less than 0.5, to factor the quadratic
  • As the frequency goes up, we hit a single pole at 1/(R₁ + R₂)C₂, as the impedance of C₂ drops, comparable to (R₁ + R₂)
  • As the frequency goes further up, impedance of C₂ drops, comparable to R R₂, the zero kicks in at 1/R₂ C₂, which levels up the gain drop to a voltage divider R₂/ (R₁ + R₂)
  • Until the frequency goes so high, now the impedance of Cs starts to show another pole at 1/(R1 ∥ R2)Cs, which drops the gain again at 20 dB/dec

Reference

David Middlebrook’s original paper

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