查看“︁双二阶滤波器”︁的源代码

'''双二阶滤波器'''的[[传递函数]]有如下的形式


<math>G(s)=\frac{p_2*s^2+p_1*s+p_0}{s^2+e_1*s+e_0}</math>

或

<math>G(s)=\frac{p_2*s^2+p_1*s+p_0}{s^2+\frac{\omega_0}{Q}*s+\omega_0^2}</math>

分子二项式中系数<math>p_2</math>,<math>p_1</math>决定滤波器的类型：
==双二阶低通滤波器==



<math>G(s)=\frac{p_0}{s^2+\frac{\omega_0}{Q}*s+\omega_0^2}</math>

其衰减函数为<ref>Adel S. Sedra, Peter O. Brackett, Filter Theory and Design, Active and Passive, p29,Matrix Publisher 1978</ref>\<ref>Rolf Schaumann,Haoqiao Xiao,Mac E. Van Valkenburg,Analog Filter Design, p144-148, Oxford University Press, 2013</ref>。

<math>A(\Omega)=G(j*\omega)*G(-j*\omega)=\frac{Q^2}{(\Omega^4*Q^2-2*\Omega^2*Q^2+\Omega^2+Q^2)}</math>

其中
<math>\Omega=\frac{\omega}{\omega_0}</math>
{{Gallery
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|File:LPcircuit2.jpg|无源双二阶[[低通滤波器]] 
|Image:BiquadFilter1.svg| 有源双二阶带通、低通滤波器电路
|File:Magnitude of low pass biquad filter.gif|对于不同的Q值，二阶低通滤波器的衰减函数曲线
|File:Phase of low pass biquad filter.png|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<ref name=as>Adel Sedra p31</ref>

<math>p_0=\frac{1}{LC}</math>

<math>\omega_0=\sqrt{\frac{1}{LC}}</math>

<math>Q=R*\sqrt{C/L}</math>

;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

<math>G(s)=\frac{s^2}{s^2+\frac{\omega_0}{Q}*s+\omega_0^2}</math>

双二阶高通滤波片的频率响应：

<math>A(\Omega)=\frac{-Q^2*\Omega^4}{(\Omega^4*Q^2-2*\Omega^2*Q^2+\Omega^2+Q^2)}</math>
{{Gallery
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|width=250
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|File:HPcircuit.jpg|无源双二阶高通滤波器
|File:Response of biqud high pass filter.gif|双二阶高通滤波器响应图
|File:Phase of high pass biquad filter.png|双二阶高通滤波器的相角
}}

<math>p_2=1</math>

<math>\omega=\frac{1}{\sqrt{LC}}</math>

<math>Q=R*\sqrt{C/L}</math>

==双二阶带通滤波器==

双二阶[[带通滤波器]]的传递函数为<ref>Adel Sedra, p26</ref>。

<math>G(s)=\frac{p_1*s}{s^2+\frac{\omega_0}{Q}*s+\omega_0^2}</math>


<math>A(\Omega)=\frac{\Omega^2*Q^2}{(\Omega^4*Q^2-2*\Omega^2*Q^2+\Omega^2+Q^2)}</math>

相角：<ref>R.Schaumann,H.Xiao and  M.Van Valkenburg, p149</ref>

<math>\theta := 90-180*arctan(\frac{\omega*\omega_0}{(Q*(\omega_0^2-\omega^2)})/\pi</math>
{{Gallery
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|File:BANDPASSCIRCUIT.jpg|双二阶无源带通滤波器
|File:Zeros and pole of biquad band pass filter.png|双二阶带通滤波器的零点和极点
|File:Frequency Response of biqud band pass filter.gif|双二阶带通滤波器频率响应
|File:Phase of band pass biquad filter.png|双二阶带通滤波器的相角
}}

<math>p_1=\frac{1}{CR}</math>

<math>\omega=\frac{1}{\sqrt{LC}}</math>

<math>Q=R*\sqrt{C/L}</math>

==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>

<math>G(s)=\frac{p_2*s^2+p_0}{s^2+\frac{\omega_0}{Q}*s+\omega_0^2}</math>

其频率响应

<math>A(\Omega)=\frac{Q^2*(\Omega^4-2*\Omega^2+1)}{(\Omega^4*Q^2-2*\Omega^2*Q^2+\Omega^2+Q^2)}</math>

相角：

<math>theta := 180*arctan(\frac{\Omega}{(Q*(\Omega^2-1)})/\pi</math>
{{Gallery
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|width=250
|height=220
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|File:Bandrejectfilter.jpg|无源双二阶带阻滤波器
|File:Response of biqud band reject filter.gif|双二阶带阻滤波器的频率响应
|File:The phase angle of band reject biquad filter.png|双二阶带阻滤波器的相角
}}
<math>p_2=1</math>

<math>\omega=\frac{1}{\sqrt{LC}}</math>

<math>Q=R*\sqrt{C/L}</math>

==参考文献==
<references/>

[[Category:电路]]