Page 219 - Anatomy of a Robot

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204 CHAPTER EIGHT
ANALOG FILTERS
One simple way to make an antialias filter is with traditional analog electronics. With
very few analog components, it’s possible to get a filter with a decent rolloff and stop-
band. Figure 8-8 shows a schematic of a simple second-order filter and the transfer
function that goes with it.
L is the inductance, C is the capacitance, and R is the resistance. Resorting to Laplace
notations for the moment, the differential equation for this circuit is derived as follows:

Vout ((1/Cs)/(1/Cs R sL))Vin
2
Vout Vin/(s LC RCs 1)
This same calculation is carried out at the following web sites:

www.ee.polyu.edu.hk/staff/eencheun/EE251_material/
Lecture1-2/lecture1-25.htm
http://engnet.anu.edu.au/courses/engn2211/notes/transnode19.html
www.engr.sjsu.edu/filt175_s01/Proj_sp2ka/act_fil_cosper_fold/act_fil_
cosper.htm
www.t-linespeakers.org/tech/filters/Sallen-Key.html

The transfer function is shown in Figure 8-9. The rolloff of this circuit is 12 db per
octave. Since this particular circuit rolls off indefinitely, the stopband should be well
below the noise floor of the input signal (and thus not a factor).
We should recognize that the differential equation of this circuit is very similar to the
second-order control system we studied in Chapter 2 on control systems. That’s because
direct analogies exist between the types of components as follows:
Capacitors are the analog of mass. Just like energy is stored in mass as it gains
speed, so, too, energy is stored in a capacitor as electrons flow into it and the volt-
age builds up.

-
L R
Vout
Vin
+
C

FIGURE 8-8 A simple second-order analog filter