Page 220 - Anatomy of a Robot

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Log Amplitude
1 DIGITAL SIGNAL PROCESSING (DSP) 205
0.5
0
Log Freq
1 2 3 4
-0.5
-1
-12 dB/octave
-1.5
-2

FIGURE 8-9 The frequency response of the second-order analog filter

Inductors are the analog of springs. Inductors, like springs, act as an energy stor-
age element. Current moves through an inductor, creates a field around the induc-
tor, and builds up the voltage across it. Just like a spring can run out of stretch, so
too an inductor can exhaust the magnetic materials that absorb energy to create
the field around the inductor. As long as the amount of energy stored in the induc-
tor stays below a certain amount, it will function properly. The same is true of a
spring.
Resistors are the analog of friction. A resistor, like friction, acts to slow down and
drain off the movement of energy between the other two components in the circuit.
The filter’s response to a step input is shown in Figure 8-10. The curve should look
very familiar since it’s virtually identical to the second-order control system we dis-
cussed before. The circuit could be used to drive a servo amplifier, but we leave it up
to the readers to figure out, given R, L, and C, how to find the values of the damping
constant d and the frequency v. It’s not our business here to use this circuit for anything
other than an antialias filter.
Given our example of a system with a 40 db S/N ratio, and using this particular cir-
cuit as an antialias filter, we can see what compromises we might have in the design of
our sampling system:
If we have a second-order analog filter with a 12 db per octave rolloff, we’d need
better than 3 octaves to attain the desired rolloff for antialiasing:

(3 octaves 12 db/octave 4 db) 40 db