Added content

filters.mdown

@@ -46,4 +46,30 @@ $$
 \end{circuitikz}
 $$
 
-$$V_o = V_i \frac{Z_{R//C}}{Z_L+Z_{R//C}} $$
+$$V_o = V_i \frac{Z_{R//C}}{Z_L+Z_{R//C}} $$
+
+**Butterworth filter**. Maximum flatness filter. The normalized Butterworth polynomials can be used to determine the transfer function for any low-pass filter cut-off frequency $\omega_c$, as follows
+
+$$H(s) = \frac{G_0}{B_n(a)}$$
+
+where $a = \frac{s}{\omega_c}$.
+
+To four decimal places, Butterworth polynomials are (expressed in quadratic factors)
+
+| **Order** | **Polynomial $B_n(s)$** |
+|-----------|----------------------------------------------------------------|
+| 1 | $(s+1)$ |
+| 2 | $s^2+1.4142s+1$ |
+| 3 | $(s+1)(s^2+s+1)$ |
+| 4 | $(s^2+0.7654s+1)(s^2+1.8478s+1)$ |
+| 5 | $(s+1)(s^2+0.6180s+1)(s^2+1.6180s+1)$ |
+| 6 | $(s^2+0.5176s+1)(s^2+1.4142s+1)(s^2+1.9319s+1)$ |
+| 7 | $(s+1)(s^2+0.4450s+1)(s^2+1.2470s+1)(s^2+1.8019s+1)$ |
+| 8 | $(s^2+0.3902s+1)(s^2+1.1111s+1)(s^2+1.6629s+1)(s^2+1.9616s+1)$ |
+
+From [Butterworth filter, Wikipedia](https://en.wikipedia.org/wiki/Butterworth_filter)
+
+
+* two frequencies which ratio is 1:10 are separate by a decade
+* two frequencies which ratio is 1:2 are separate by a octave
+* 1 decade is 3.33 octaves