John:
[snip]
we make an active filter out of 2nd-order unilateral sections. Since
interactions are reduced, we have another degree of freedom in the
design.
So why does the presence of the buffer increase component sensitivity?
If you can explain why it does, I'll be suitably embarassed. [1]
John
[snip]
Actually it is just those interactions that you are trying to get rid of
that make
the lossless LC ladder so insensitive.
To see that a properly doubly terminated lossless LC ladder filter, which
has
all of those "intereactions" has the absolute lowest element value
sensitivities, i.e.
the sensitivities are *zero*, consider the following:
A lossless Nth order LC network optimally matched [conjugate matched] at
its passband reflection zeros has an attenuation A(f) of exactly zero at
those
reflection zeros "frk". i.e. the attenuation A(frk) = 0 dB at those N
passband
frequencies frk. Now apply a signal at any one of those frequencies of
zero attenuation, say fr and measure the attenuation at that frequency
A(fr, Ln, Cm) = 0
Then vary any Ln or Cm component. Since the filter is lossless, [L-C] there
is
no power lost in the filter and so the attenuation A simply cannot
increase, it
can only decrease from zero.
Now plot A(fr, Ln, Cm) versus any L or C value as it deviates from it's
nominal
design value. Pot attenuation A(fr) in dB on the vertical scale and say C
in Farads
for any capacitor, or inductor, across the horizontal scale.
The curve you measure of A versus C clearly touches zero attenuation
at the nominal value of C and then only increases as C is varied above and
below
its nominal value. The lossless network can only insert more loss it can't
create
gain!
The curve that you plot in this way is the sensitivity of the design to that
C.
Clearly this sensitivity has a minimum value, saddle point, at the nominal
design
value and that sensitivity function has the minimum *zero*.
You can't get much lower than that my friend!
This is true at every reflection zero in the passband and very nearly so
everywhere
throughout the passband if the passband loss is small.
The heuristic physical reasoning that I just shared with you has a rigorous
mathematical
proof, and there are some very neat simple formulas allowing you to
pre-compute
the maximum value of the sensitivity function before you even start the
design.
The matched lossless LC ladder filter operating between matched resistive
terminations
is the least sensitive and most robust filter design known to man!
Same is true even for active RC and digitial filters. The least sensitive
digital filters, least
sensitive to round off and other numerical errors and artifacts that is, are
those digital filter
structures that "mimic" lossless LC ladders.
What more can I tell you.
If you are faced with high precision demanding stringent robust design
requirments for filtering
then you can do no better than to use matched lossless LC filter design or
something that mimics
it's "interactions".
Don't eliminate "interactions" as you suggest, make good use of them.
Well designed complex interactons are what make good filter design good!
