9 Filter Tables

Typically, filter books list the zeros or the coefficients of the particular polynomial being used to define the filter type. As seen previously in this document, a certain amount of mathematical manipulation is required to turn this information into a circuit realization. The relationships between the zeros and the frequency scaling factor ( $FSF$ ) and quality factor ( $Q$ ) are given by $FSF=\sqrt{{\mathrm{Re}}^2+{\left|\mathrm{lm}\right|}^2}$ and $Q=\frac{\sqrt{{\mathrm{Re}}^2+{\left|\mathrm{lm}\right|}^2}}{2\mathrm{Re}}$ , where $\mathrm{Re}$ and $\mathrm{Im}$ are the real and imaginary parts of the complex-conjugate zero pair. Table 9-1 through Table 9-4 are generated in this way. Higher-order filters are constructed by cascading second-order stages for even-order filters (one for each complex-conjugate zero pair). A first-order stage is then added if the filter order is odd. With the filter tables arranged this way, the preliminary mathematical work is done and the designer is only left with calculating the circuit component values based on three formulas.

For a low-pass Sallen-Key filter with cutoff frequency $f_{\mathrm{c}}$ and pass-band gain $K$ , set $K=\frac{{\mathrm{R}}_3+{\mathrm{R}}_4}{{\mathrm{R}}_3}$ , $FSF\times f_{\mathrm{c}}=\frac{1}{2\mathrm{\pi }\sqrt{{\mathrm{R}}_1{\mathrm{R}}_2{\mathrm{C}}_1{\mathrm{C}}_2}}$ , and $Q=\frac{\sqrt{{\mathrm{R}}_1{\mathrm{R}}_2{\mathrm{C}}_1{\mathrm{C}}_2}}{{\mathrm{R}}_1{\mathrm{C}}_1+{\mathrm{R}}_2{\mathrm{C}}_1+{\mathrm{R}}_1{\mathrm{C}}_2(1-K)}$ for each second-order stage. If an odd order is required, set $FSF\times f_{\mathrm{c}}=\frac{1}{2\mathrm{\pi RC}}$ for that stage.

For a low-pass MFB filter with cutoff frequency $f_{\mathrm{c}}$ and pass-band gain $K$ , set $K=\frac{-{\mathrm{R}}_2}{{\mathrm{R}}_1}$ , $FSF\times f_{\mathrm{c}}=\frac{1}{2\mathrm{\pi }\sqrt{{\mathrm{R}}_2{\mathrm{R}}_3{\mathrm{C}}_1{\mathrm{C}}_2}}$ , and $Q=\frac{\sqrt{{\mathrm{R}}_2{\mathrm{R}}_3{\mathrm{C}}_1{\mathrm{C}}_2}}{{\mathrm{R}}_3{\mathrm{C}}_1+{\mathrm{R}}_2{\mathrm{C}}_1+{\mathrm{R}}_3{\mathrm{C}}_1(-K)}$ for each second-order stage. If an odd order is required, set $FSF\times f_{\mathrm{c}}=\frac{1}{2\mathrm{\pi RC}}$ for that stage.

The tables are arranged so that increasing $Q$ is associated with increasing stage order. High-order filters are normally arranged in this manner to help prevent clipping.