double feedback delta^2 mod, but also lowpass 100kHz with input

circuit-20251127-0040.svg

sim 10MHz clock, no input, oscillates around 500 kHz

sim with input

The equation is a sum of double integral and a single integral

Vout=
$$\frac{1}{(RC)^2} \int\int Q(t)dt + \frac{1}{RC} \int Q(t) dt$$

But the double integral dominates, because it is over the square of RC...and since RC < 1, that means 1/(RC)^2 >> 1/(RC).

sim again, but for 5 ms

Q output before integrations:

sim 100Mhz without flipflop, oscillates around 1Mhz

before integration:

try using 20kHz cutoff, this lowers oscillation to around 230 kHz

very clean:

adding that final 20kHz pole seems to do something interesting to the overall phase shift:

The 230kHz oscillation frequency seems to correspond to the rightmost phase intersection with 0 degrees, not the leftmost. Berkhausen just says 0 degrees, but doesn't say if there are multiple intersections.

since the feedback is dominated by double-integral, then maybe slope overload isn't as big as an issue, because the double integral grows like x^2 if provided a step input.

if hypothetically use infinite bandwidth opamps, then the bode plot is a constant -40dB/decade over all frequencies, and the phase reaches 0degrees around 1Mhz, so the oscillation frequnecies jumps randomly around above 1 MHz

circuitjs simulation which adds 100ns delay buffer seems to cause oscillation to drop down to just 69kHz.

retry simulate with 50ns delay buffer, lowers oscillation from 230kHz to 100kHz

very good noise behavior:

sim with 20Mhz clock

with 0 input:

with input:

oscillations around 160kHz:

test with comparator and offset error and no input

now try with input:

It seems offset error isn't really a big deal at all for this second derivative circuit...

same circuit but without any offset error:

So it seems maybe no problem at all.

if add 10Mohm feedback resistors

if add offset error now:

So although maybe adds a DC offset to result, it doesn't seem to really make much error in the audio frequency range.

video using 20kHz 1.5Vpp sine using breadboard setup with cheap components (LM393, MCP6004, SN74AC74, schmitt clock). Reconstruction does overshoot rails.

Screencast_20251128_022657.webm

2Vpp sine, so not overshooting, and don't draw on top of eachother:

Screencast_20251128_023300.webm