User:Dicklyon/filter

Filter with series impedances A and C (typically resistors, or resistors plus inductors), shunt impedances to ground B and D (typically capacitors). See my book Fig. 6.12.

Input impedance:
 * $$ Z = A + \frac{B (C+D)}{B + C + D}$$

Transfer function to after first series impedance A via voltage-divider formula:
 * $$ \frac{V2}{V1} = \frac{\frac{B (C+D)}{B + C + D}}{A + \frac{B (C+D)}{B + C + D}} $$
 * $$                      = \frac{B (C+D)}{A({B + C + D}) + B (C+D)} $$

Transfer fuction from that node to output node is just the C and D voltage divider:
 * $$ \frac{V3}{V2} = \frac{D}{C+D}$$

RC example: use $$A = C = R$$, and $$B = D = \frac{1}{sC}$$, and $$\tau = RC$$. Then:
 * $$ \frac{V2}{V1} = \frac{\frac{1}{sC} (R+\frac{1}{sC})}{R({\frac{1}{sC} + R + \frac{1}{sC}}) + \frac{1}{sC} (R+\frac{1}{sC})} $$
 * $$ = \frac{\tau s +1} {\tau s (\tau s +2) + \tau s +1} $$
 * $$ = \frac{\tau s +1} {(\tau s)^2 + 3\tau s + 1} $$


 * $$ \frac{V3}{V2} = \frac{\frac{1}{sC}}{R+\frac{1}{sC}}$$
 * $$  = \frac{1}{\tau s +1}$$


 * $$ \frac{V3}{V1} = \frac{V2}{V1} \frac{V3}{V2} = \frac{1}  {(\tau s)^2 + 3\tau s + 1} $$
 * $$  = \frac{1}{(\tau s - r_1)(\tau s - r_2)}$$

for $$r_i$$ being the roots of denominator $${(\tau s)^2 + 3\tau s + 1} $$:
 * $$r_i = \frac{-3 \pm \sqrt{3^2 - 4}}{2}$$
 * $$= -1.5 \pm 1.1180 $$
 * $$r_1 = -2.6180$$
 * $$r_2 = -0.320$$

These are the poles as normalized by $$\tau$$. Convert to absolute locations in s:


 * $$ H(s) = \frac{V3}{V1} = \frac{\tau^{-2}}  {s^2 + \frac{3s}{\tau} + \frac{1}{\tau^2}} $$
 * $$  = \frac{\tau^{-2}}{(s - p_1)(s - p_2)}$$
 * $$p_1 = \frac{-2.6180}{\tau}$$
 * $$p_2 = \frac{-0.3820}{\tau}$$

Example: $$R = 10^3, C = 10^{-6}, \tau = 10^-3$$ (1 kilohm resistors, 1 microfarad capacitors, 1 ms RC time constant):
 * $$p_1 = -2618.0$$
 * $$p_2 = -382.0$$

Find DC gain by setting $$ s = 0 $$
 * $$H(0) = \frac{\tau^{-2}}{(-p_1)(-p_2)} = \frac{10^6}{2618.0 \cdot 382.0} = 1.0$$ as expected.

Radian corner frequencies:
 * $$\omega_1 = 2618.0$$ radian/s
 * $$\omega_2 = 382.0$$ radian/s

Divide by $$-2\pi$$ to get corner frequencies in cycles per second, for making Bode plot:
 * $$f_1 = 416.7$$ Hz – high-frequency corner where transfer function magnitude changes from -6 to -12 dB/octave.
 * $$f_2 = 60.8$$ Hz – low-frequency corner where transfer function magnitude changes from flat to -6 dB/octave.

(I should have numbered 1 and 2 in the other order.)