# 双二阶滤波器

${\displaystyle G(s)={\frac {p_{2}*s^{2}+p_{1}*s+p_{0}}{s^{2}+e_{1}*s+e_{0}}}}$

${\displaystyle G(s)={\frac {p_{2}*s^{2}+p_{1}*s+p_{0}}{s^{2}+{\frac {\omega _{0}}{Q}}*s+\omega _{0}^{2}}}}$

## 双二阶低通滤波器

${\displaystyle G(s)={\frac {p_{0}}{s^{2}+{\frac {\omega _{0}}{Q}}*s+\omega _{0}^{2}}}}$

${\displaystyle A(\Omega )=G(j*\omega )*G(-j*\omega )={\frac {Q^{2}}{(\Omega ^{4}*Q^{2}-2*\Omega ^{2}*Q^{2}+\Omega ^{2}+Q^{2})}}}$

${\displaystyle p_{0}={\frac {1}{LC}}}$

${\displaystyle \omega _{0}={\sqrt {\frac {1}{LC}}}}$

${\displaystyle Q=R*{\sqrt {C/L}}}$

## 双二阶高通滤波器

${\displaystyle G(s)={\frac {s^{2}}{s^{2}+{\frac {\omega _{0}}{Q}}*s+\omega _{0}^{2}}}}$

${\displaystyle A(\Omega )={\frac {-Q^{2}*\Omega ^{4}}{(\Omega ^{4}*Q^{2}-2*\Omega ^{2}*Q^{2}+\Omega ^{2}+Q^{2})}}}$

${\displaystyle p_{2}=1}$

${\displaystyle \omega ={\frac {1}{\sqrt {LC}}}}$

${\displaystyle Q=R*{\sqrt {C/L}}}$

## 双二阶带通滤波器

${\displaystyle G(s)={\frac {p_{1}*s}{s^{2}+{\frac {\omega _{0}}{Q}}*s+\omega _{0}^{2}}}}$

${\displaystyle A(\Omega )={\frac {\Omega ^{2}*Q^{2}}{(\Omega ^{4}*Q^{2}-2*\Omega ^{2}*Q^{2}+\Omega ^{2}+Q^{2})}}}$

${\displaystyle \theta :=90-180*arctan({\frac {\omega *\omega _{0}}{(Q*(\omega _{0}^{2}-\omega ^{2})}})/\pi }$

${\displaystyle p_{1}={\frac {1}{CR}}}$

${\displaystyle \omega ={\frac {1}{\sqrt {LC}}}}$

${\displaystyle Q=R*{\sqrt {C/L}}}$

## 双二阶带阻滤波器

${\displaystyle G(s)={\frac {p_{2}*s^{2}+p_{0}}{s^{2}+{\frac {\omega _{0}}{Q}}*s+\omega _{0}^{2}}}}$

${\displaystyle A(\Omega )={\frac {Q^{2}*(\Omega ^{4}-2*\Omega ^{2}+1)}{(\Omega ^{4}*Q^{2}-2*\Omega ^{2}*Q^{2}+\Omega ^{2}+Q^{2})}}}$

${\displaystyle theta:=180*arctan({\frac {\Omega }{(Q*(\Omega ^{2}-1)}})/\pi }$

${\displaystyle p_{2}=1}$

${\displaystyle \omega ={\frac {1}{\sqrt {LC}}}}$

${\displaystyle Q=R*{\sqrt {C/L}}}$

## 参考文献

1. ^ Adel S. Sedra, Peter O. Brackett, Filter Theory and Design, Active and Passive, p29,Matrix Publisher 1978
2. ^ Rolf Schaumann,Haoqiao Xiao,Mac E. Van Valkenburg,Analog Filter Design, p144-148, Oxford University Press, 2013
3. ^ Adel Sedra p31
4. ^ Adel Sedra, p26
5. ^ R.Schaumann,H.Xiao and M.Van Valkenburg, p149