Higher order (2n) VSVCUnit GainThe Butterworth low-pass filter design, which can be decomposed into n second-order low-passes, can be optimized by combining and optimizing these multiple second-order low-passes to improve the low-pass characteristics and stability of the filter.
The transfer function of the series is the product of the transfer functions of the individual second-order filters:\({{\rm{H}}_{2n}}(s) = \prod\nolimits_{i - 1}^n {{H_2}^{(i)}(s)}\)；
Second order voltage controlled voltage source low pass filter circuit diagram:
The transfer function is obtained by "virtual short - virtual break":\(H(s) = {{\mathop V\nolimits_o } \over {\mathop V\nolimits_i }} = {{\mathop A\nolimits_F /\mathop R\nolimits_1 \mathop R\nolimits_2 \mathop C\nolimits_1 \mathop C\nolimits_2 } \over {\mathop s\nolimits^2 + s({1 \over {\mathop R\nolimits_1 \mathop C\nolimits_1 }} + {1 \over {\mathop R\nolimits_2 \mathop C\nolimits_1 }} + {{1 - \mathop A\nolimits_F } \over {\mathop R\nolimits_2 \mathop C\nolimits_2 }}) + {1 \over {\mathop R\nolimits_1 \mathop C\nolimits_1 \mathop R\nolimits_2 \mathop C\nolimits_2 }}}}\)；
included among these\(s = j\omega\)，\(\mathop A\nolimits_F = 1 + {{\mathop R\nolimits_f } \over {\mathop R\nolimits_r }}\)；
De-normalize the transfer function of the low-pass filter:\(H(s) = {{\mathop H\nolimits_0 \mathop \omega \nolimits_0^2 } \over {\mathop S\nolimits^2 + \alpha \mathop \omega \nolimits_0 S + \beta \mathop \omega \nolimits_0^2 }}\)；
included among these\(\beta \mathop \omega \nolimits_0^2 = {1 \over {\mathop R\nolimits_1 \mathop R\nolimits_2 \mathop C\nolimits_1 \mathop C\nolimits_2 }}\)，\(\mathop H\nolimits_0 \mathop \omega \nolimits_0^2 = {{\mathop A\nolimits_F } \over {\mathop R\nolimits_1 \mathop R\nolimits_2 \mathop C\nolimits_1 \mathop C\nolimits_2 }}\)，\(\alpha \mathop \omega \nolimits_0 = {1 \over {\mathop R\nolimits_1 \mathop C\nolimits_1 }} + {1 \over {\mathop R\nolimits_2 \mathop C\nolimits_1 }} + {{1 - \mathop A\nolimits_F } \over {\mathop R\nolimits_2 \mathop C\nolimits_2 }}\)；
\({\omega _0}\)is the cutoff angular frequency.\(\alpha\)、\(\beta\)are binomial coefficients representing different filtering characteristics.
preferences\(\mathop C\nolimits_2 = k\mathop C\nolimits_1\)So.\(\mathop H\nolimits_0 = \beta \mathop A\nolimits_F\)，\(\beta \mathop k\nolimits^2 \mathop \omega \nolimits_0^2 \mathop C\nolimits_1^2 \mathop R\nolimits_2^2 - \alpha k\mathop \omega \nolimits_0 \mathop C\nolimits_1 \mathop R\nolimits_2 + (1 + k - \mathop A\nolimits_F ) = 0\)(on\({R_2}\)(the quadratic equation), since\({R_2}\)There exists a real number solution, then k must satisfy\(k \le {{\mathop \alpha \nolimits^2 } \over {4\beta }} + \mathop A\nolimits_F - 1\);
The solution can be obtained:\(\mathop R\nolimits_1 = {{\alpha \mp \sqrt {{\alpha ^2} - 4\beta (1 + k - {A_F})} } \over {2\beta (1 + \kappa - {{\rm A}_F}){\omega _0}{C_1}}}\)，\(\mathop R\nolimits_2 = {{\alpha \pm \sqrt {{\alpha ^2} - 4\beta (1 + k - {A_F})} } \over {2\beta k{\omega _0}{C_1}}}\)
chosen and fixed\({C_1}\), k and then the VSVC low-pass filter with arbitrary characteristics is designed according to the calculation formula.
Normalized Butterworth polynomials:
For unit gain\(\mathop A\nolimits_F = 1\), second-order low-pass, polynomial coefficients\(\beta=1\)；
in that case\(\mathop H\nolimits_0 = 1\)，\(k \le 0.25{\alpha ^2}\)(k takes the value\(0.25{\alpha ^2}\)VCVS second-order unit-gain low-pass offers the advantages of convenience, low cost and stability at the same time) and\(\mathop R\nolimits_1 = {{\alpha \mp \sqrt {{\alpha ^2} - 4k} } \over {2k{\omega _0}{C_1}}}\)，\(\mathop R\nolimits_2 = {{\alpha \pm \sqrt {{\alpha ^2} - 4k} } \over {2k{\omega _0}{C_1}}}\)。
Typically, it is convenient to design hardware circuits that make the\({R_1} = {R_2}\)。\({C_1}\)The selection is generally based on the empirical formula\({C_1} \approx {10^{ - 3 \sim - 5}}{f_0}^{ - 1}\)Derived.
This further simplifies to:\({C_2} = 0.25{\alpha ^2}{C_1}\)，\({R_1} = {R_2} = {2 \over {\alpha {\omega _0}{C_1}}} = {1 \over {\pi \alpha {f_0}{C_1}}}\)。
In addition, a loop is provided for the positive end of the op-amp to compensate for the misalignment, which is taken as\({R_f} \ll {R_r},{R_f}//{R_r} \approx {R_f} = {R_1} + {R_2} = {2 \over {\pi \alpha {f_0}{C_1}}}\)This completes the parameterization of the low-pass second-order Butterworth low-pass filter.
For higher order LPF designs, this is accomplished by referring to the polynomial coefficients and the set cutoff frequency.
Example simulation design:A fourth-order Butterworth low-pass filter is designed with a cutoff frequency of 100khz and a gain of one:
Fourth-order low-pass presence parameters:\({\alpha _1} = 0.7654,{\alpha _2} = 1.8478\), f=100khz, take the first stage\second stage\({C_1} = 4.7nF\)；
Get:
first level\({C_2} = 0.68nF\)，\({R_1} = {R_2} = 884.8Ω\)，\({R_f} = 1769.6Ω\)；
second level\({C_2} = 4.02nF\)，\({R_1} = {R_2} = 366.5Ω\)，\({R_f} = 733Ω\)，
\({R_r}\)Take 1MΩ. multisim simulation is as follows:
