This looks right dimensionally, and it goes to V/I = C1 R3 R4 s as the GBW
product (GBW = k / (2 * pi)) goes to infinity. So I think I have my head
screwed on straight.
Dinner was recent and rich, and intuition is coming slowly to me, but if
you assume that C1 R4 << k, then the first thing that will show up as
frequency increases is the
s + k/2
----------
s + k
part. That's going to make the impedance lead even more than just an
inductance, which is oddball. I'm pretty sure that'll appear as an
inductor with some negative resistance, which will artificially narrow up
any tank circuit you throw the gyrator into (or make it oscillate, if the
Q and frequency are high enough). I don't think that the dynamics of C1
and R4 will contribute much unless you let C1 R4 approach k -- and dangit,
I'm not going to figure out if doing so makes things better or worse;
that's your job!
It looks like it'll have a seriously underdamped pole pair with a damping
rate of k/8 and a natural frequency that roughly depends on sqrt(k/
(2*C1*R4)). Whether this causes problems or not -- I dunno. As long as
the amplifier gain is strictly equal to k/s then it's unconditionally
stable, and I think that a single additional pole (A = k / (s * (s + a)))
or other equivalent lag would actually make it more stable, not less. I
wouldn't bet on that stability unless I actually did some paying work on
it (and besides, you can do that part).
(BTW: that last paragraph came completely from sketching a root locus
diagram and calculating the intersection of the poles-to-infinity locus
with the real line -- I think that's the first time I've actually done
the intersection thing to predict the behavior of a real live system, and
not just as a homework exercise for a class. So -- 28 years after I took
the class, I actually used that tidbit for more than getting a grade).
