双二阶滤波器

双二阶滤波器的传递函数有如下的形式

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

或

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

分子二项式中系数${\displaystyle p_{2}}$,${\displaystyle p_{1}}$决定滤波器的类型：

双二阶低通滤波器

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

其衰减函数为^[1]\^[2]。

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

其中 ${\displaystyle \Omega ={\frac {\omega }{\omega _{0}}}}$ 

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  无源双二阶低通滤波器
•  
  有源双二阶带通、低通滤波器电路
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  对于不同的Q值，二阶低通滤波器的衰减函数曲线
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  双二阶低通滤波器的相角

无源双二阶低通滤波器

无源双二阶低通滤波器由电阻、电容和电感元件组成^[3]

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

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

${\displaystyle Q=R*{\sqrt {C/L}}}$ 

有源双二阶低通滤波器

有源双二阶低通滤波器由运算放大器、电容、电感和电阻构成。

双二阶高通滤波器

双二阶高通滤波器的传递函数为

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

双二阶高通滤波片的频率响应：

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

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  无源双二阶高通滤波器
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  双二阶高通滤波器响应图
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  双二阶高通滤波器的相角

${\displaystyle p_{2}=1}$ 

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

${\displaystyle Q=R*{\sqrt {C/L}}}$ 

双二阶带通滤波器

双二阶带通滤波器的传递函数为^[4]。

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

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

相角：^[5]

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

•  
  双二阶无源带通滤波器
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  双二阶带通滤波器的零点和极点
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  双二阶带通滤波器频率响应
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  双二阶带通滤波器的相角

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

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

${\displaystyle Q=R*{\sqrt {C/L}}}$ 

双二阶带阻滤波器

双二阶带阻滤波器的传递函数为<refnname=rs>Rolf Schaumann,H.Xiao,M.E.van Valkenburg, p225</ref>

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

其频率响应

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

相角：

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

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  无源双二阶带阻滤波器
•  
  双二阶带阻滤波器的频率响应
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  双二阶带阻滤波器的相角

${\displaystyle p_{2}=1}$ 

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

${\displaystyle Q=R*{\sqrt {C/L}}}$ 

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