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Wed Dec 22 17:01:23 EST 2010

Capacitors and resistors: getting a feel for component values

Managing Impedance
------------------

One of the essential elements of good circuit design is to manage the
orders of magnitude -- to optimize circuit impedance.

Most designs allow to be run at higher current/voltage which is good
for noise immunity but bad for component cost (bigger size) and
operating cost (larger power cost, heat production).

I remember this to be one of the earliest discouraging problems I ran
into, trying to obtain expensive large Polypropylene capacitors for a
filter circuit I designed on paper because the resistors values I used
where on the small side.  In those days I had nobody to turn to and no
access to information that was in a form I could digest, so I gave up.

In recent years and especially the last couple of weeks getting my
hands dirty and reading schematics, these things have become more
clear to me.  So, as an illustration, here's a problem I ran into
which hints at the "muscle memory" you need to create about resistor
and capacitor values.


Dynamic range
-------------

The problem is that both resistors and caps have a large dynamic range
of useful values -- about 8 decades each.

  - 0.1   (10^-1) -> 10M (10^7):
  - 1000u (10^-3) -> 10p (10^-11)

As a result the time constants that come from combining these have
about 15 decades.  That's a pretty large time scale from 11 days to 1
picosecond (1 teraHz).

The time scales for audio processing are luckily a bit less large,
about 3 decades from 20Hz to 20kHz, or 4 from 5-50kHz depending on how
you count.

However, due to the large range of R and C, there are about 10 - 11
decades of impedance to choose from when RC time constants are given.
That's quite a lot.  Most of this is ruled out as either too high
(instrumentation amplifier) or too low (power amp), which leaves a
good 3-4 decades where resistors range from 220Ohm to 2M.

The Art of Electronics[1] (p. 279) has a rule of thumb that that says
to pick a capacitor value in based of 10uF / Hz.  This places resistor
values around 10-20k.  It doesn't really explain why.  At another
point ([1] p. 275) it mentions to pick resistor values in the 10k-100k
range, as lower values tend to get closer to the open loop impedance
of opamps as frequencies rise.  My guess is that higher resitor values
are to be avoided because of noise. (TODO: add some quantitative
explanation here).


The problem
-----------

Currently I have only 50k dual log (A-type) potentiometers I want to
use in a SVF filter.  The lowest frequency of interest is 20Hz, which
brings the capacitor value to about 160nF, which rounds down
(frequency) to 220nF.

That seems rather large for a ceramic cap.  The largest I have here is
100nF, and it's a ceramic multilayer (the microphonic ones).  The
smallest ceramic disc cap I have is 47nF (473).

Let's check some Futurlec[2] prices to see where they start to rise
significantly.

Capacitance  Max     Smaller  Price   Largest   Type
-----------------------------------------------------------------------
100nF        2200nF  $0.07    $0.12   $0.60     100V Mylar
150nF        2200nF  $0.06    $0.15   $0.30     50V Multilayer Ceramic
100nF         100nF  $0.05    $0.10   $0.10     50V Ceramic Disc

That looks like a trend.  From the cost perspective it seems to be a
good idea to keep capacitors below 100nF, which means impedances
should be 100k or above for to get to RC frequencies below 20Hz.



Conclusions
-----------

For full audio range SVF time constant setting, use a 220n cap if 50k
is given, but if possible, raise the POT to 500k and use a 22n cap.

EDIT: For SVF, the loop gain can be lowered to end up with smaller cap
values for the same frequency range.



[1] isbn://0521370957
[2] http://www.futurlec.com/Capacitors.shtml








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