双二阶滤波器

双二阶滤波器的传递函数有如下的形式 G ( s ) = 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}+e_{1}*s+e_{0}}}} 或 G ( s ) = p 2 ∗ s 2 + p 1 ∗ s + p 0 s 2 + ω 0 Q ∗ s + ω 0 2 {\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 p 2 {\displaystyle p_{2}} , p 1 {\displaystyle p_{1}} 决定滤波器的类型： G ( s ) = p 0 s 2 + ω 0 Q ∗ s + ω 0 2 {\displaystyle G(s)={\frac {p_{0}}{s^{2}+{\frac {\omega _{0}}{Q}}*s+\omega _{0}^{2}}}} 其衰减函数为\。 A ( Ω ) = G ( j ∗ ω ) ∗ G ( − j ∗ ω ) = Q 2 ( Ω 4 ∗ Q 2 − 2 ∗ Ω 2 ∗ Q 2 + Ω 2 + Q 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 Ω = ω ω 0 {\displaystyle \Omega ={\frac {\omega }{\omega _{0}}}} 无源双二阶低通滤波器 有源双二阶带通、低通滤波器电路 对于不同的Q值，二阶低通滤波器的衰减函数曲线 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 p 0 = 1 L C {\displaystyle p_{0}={\frac {1}{LC}}} ω 0 = 1 L C {\displaystyle \omega _{0}={\sqrt {\frac {1}{LC}}}} Q = R ∗ C / L {\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。 双二阶高通滤波器的传递函数为 G ( s ) = s 2 s 2 + ω 0 Q ∗ s + ω 0 2 {\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： A ( Ω ) = − Q 2 ∗ Ω 4 ( Ω 4 ∗ Q 2 − 2 ∗ Ω 2 ∗ Q 2 + Ω 2 + Q 2 ) {\displaystyle A(\Omega )={\frac {-Q^{2}*\Omega ^{4}}{(\Omega ^{4}*Q^{2}-2*\Omega ^{2}*Q^{2}+\Omega ^{2}+Q^{2})}}} 无源双二阶高通滤波器 双二阶高通滤波器响应图 双二阶高通滤波器的相角 p 2 = 1 {\displaystyle p_{2}=1} ω = 1 L C {\displaystyle \omega ={\frac {1}{\sqrt {LC}}}} Q = R ∗ C / L {\displaystyle Q=R*{\sqrt {C/L}}} 双二阶带通滤波器的传递函数为。 G ( s ) = p 1 ∗ s s 2 + ω 0 Q ∗ s + ω 0 2 {\displaystyle G(s)={\frac {p_{1}*s}{s^{2}+{\frac {\omega _{0}}{Q}}*s+\omega _{0}^{2}}}} A ( Ω ) = Ω 2 ∗ Q 2 ( Ω 4 ∗ Q 2 − 2 ∗ Ω 2 ∗ Q 2 + Ω 2 + Q 2 ) {\displaystyle A(\Omega )={\frac {\Omega ^{2}*Q^{2}}{(\Omega ^{4}*Q^{2}-2*\Omega ^{2}*Q^{2}+\Omega ^{2}+Q^{2})}}} 相角： θ := 90 − 180 ∗ a r c t a n ( ω ∗ ω 0 ( Q ∗ ( ω 0 2 − ω 2 ) ) / π {\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 双二阶带通滤波器的零点和极点 双二阶带通滤波器频率响应 双二阶带通滤波器的相角 p 1 = 1 C R {\displaystyle p_{1}={\frac {1}{CR}}} ω = 1 L C {\displaystyle \omega ={\frac {1}{\sqrt {LC}}}} Q = R ∗ C / L {\displaystyle Q=R*{\sqrt {C/L}}} 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> G ( s ) = p 2 ∗ s 2 + p 0 s 2 + ω 0 Q ∗ s + ω 0 2 {\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 A ( Ω ) = Q 2 ∗ ( Ω 4 − 2 ∗ Ω 2 + 1 ) ( Ω 4 ∗ Q 2 − 2 ∗ Ω 2 ∗ Q 2 + Ω 2 + Q 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})}}} 相角： t h e t a := 180 ∗ a r c t a n ( Ω ( Q ∗ ( Ω 2 − 1 ) ) / π {\displaystyle theta:=180*arctan({\frac {\Omega }{(Q*(\Omega ^{2}-1)}})/\pi } 无源双二阶带阻滤波器 双二阶带阻滤波器的频率响应 双二阶带阻滤波器的相角 p 2 = 1 {\displaystyle p_{2}=1} ω = 1 L C {\displaystyle \omega ={\frac {1}{\sqrt {LC}}}} Q = R ∗ C / L {\displaystyle Q=R*{\sqrt {C/L}}} Adel S. Sedra, Peter O. Brackett, Filter Theory and Design, Active and Passive, p29,Matrix Publisher 1978 Rolf Schaumann,Haoqiao Xiao,Mac E. Van Valkenburg,Analog Filter Design, p144-148, Oxford University Press, 2013 Adel Sedra p31 Adel Sedra, p26 R.Schaumann,H.Xiao and M.Van Valkenburg, p149