filters.mdown

% Filters %Diego Trapero

Filters

LC filter

LC filter loaded with R transfer function

$$ \begin{circuitikz}[scale = 1] \draw (0,2) nodeocirc{} (vi) node[left] {$v_i$} (4,2) nodeocirc{} (vo) node[right] {$v_o$} (vi) toL, l=$L$ (2,2) toC, l=$C$ (2,0) nodeground{} (3,2) toR, l=$R$ (3,0) nodeground{} (2,2) toshort ; \end{circuitikz} $$

In the voltage-divider form:

$$ \begin{circuitikz}[scale = 1] \draw (2,4) nodeocirc{} (vi) node[left] {$v_i$} (4,2) nodeocirc{} (vo) node[right] {$v_o$} (vi) toL, l=$L$ (2,2) toC, l=$C$ (2,0) nodeground{} (3,2) toR, l=$R$ (3,0) nodeground{} (2,2) toshort ; \end{circuitikz} $$

$$V_o = V_i \frac{Z_{R//C}}{Z_L+Z_{R//C}} $$

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

• 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

$$\frac{6dB}{oct} = \frac{20dB}{dec}$$