Bro and the Ph.D in Earth Field

Same picnic table, one year later. 
The epoxy “cuddle-blob” is now under a machined-aluminium heat-spreader anodised matte-black.  
 
Bro’s coffee cup reads “∂V/∂t keeps me awake.”  
 
Ph.D’s cup still warns about δT/δt, but the font is now Comic Sans—an inside joke about peer-review formality.  
 
They open their laptops side-by-side; the RP2040 has been promoted to a dual-core STM32H7, yet the $20 budget somehow still balances.]

BRO (M.S. EE, still wearing skate shoes):

I’ve been modelling the auto-zero loop as a discrete-time Lyapunov drift.  
 
If I treat the digi-pot wiper position as state xₖ, then ∆xₖ = −µ·sign(Vos,ₖ) gives global asymptotic stability for 0 < µ ≤ 1 LSB.  
 
But the residual dithers between ±½ LSB.  I’m thinking… inject a shaped PRBS so the error spectrum spreads like spread-spectrum CDMA.  
 
That way we gain 3 dB effective resolution.  Legit?

EE Ph.D (raising eyebrow, pleased):

Legit.  You’re basically dithering a 1-bit ΣΔ inside a multi-bit wrapper.  
 
Keep the PRBS amplitude at ¼ LSB so the peak error stays inside the bound, and high-pass above the 1/f corner—your 10 Hz Schumann line stays untouched.  
 
Nice bridge between stochastic control and classic mixed-signal lore.

BRO (grinning, but calm):

Cool.  I’ll call it “whisper-noise dither” so my homies don’t fall asleep.  
 
Skate-video soundtrack meets Shannon—perfect.

BRO:

While I was at it, I re-did the tail-current source.  
 
Instead of a lone 33 kΩ, I stacked two 0.1 % thin-films with a 2.5 V reference and a BJT current mirror.  
 
Temperature coefficient dropped to 8 ppm/°C.  
 
We’re now burning only 80 µA, yet output impedance hit 2 MΩ at 10 Hz.  
 
That’s… spa-day quiet.

EE:

You just described a Brokaw-bandgap-less reference topology, minus the bandgap.  
 
Clever—by letting the mirror run at < 0.5 V VCE you avoid the curvature term.  
 
You realise you’ve entered the “sub-threshold, sub-ppm” club?  Membership includes log amplifiers and seismic front-ends.

BRO:

Membership card can double as a guitar pick—multifunctional, like us.

BRO (pulling up LTspice):

Long-tailed pair AC simulation shows CMRR peaking at 94 dB @ 100 Hz, then rolling 20 dB/dec above 3 kHz because of the 2 pF protection diodes.  
 
I’m eyeing bootstrapping the guard ring with a miniature op-amp to keep the diodes at zero bias.  
 
Think we can push CMRR > 120 dB out to 10 kHz?

EE:

Theoretically, yes—bootstrap cancels the junction capacitance, but watch the phase-margin.  
 
You’re introducing a feed-forward path; loop gain may kiss unity at 2 MHz and sing.  
 
Add a 10 Ω damping resistor inside the boot loop, keeps Q < 0.5.  
 
Also, mind the guard-ring leakage: 1 GΩ || 1 pF creates a zero at 160 Hz that’ll eat your lunch.

BRO (nodding):

Trade-offs, always trade-offs.  I’ll 3-D print a Teflon shield so the guard never sees sweat—literally and metaphorically.

BRO (leaning back, eyes on spectrum analyser):

I ported the interval-to-MIDI code to Rust, zero-copy, runs in 120 µs on the M7.  
 
Latency jitter is now 400 ns RMS—smaller than the crystal period.  
 
We could clock it from a TCXO and stream timestamps over 100BASE-TX.  
 
Lab-grade time transfer, still field-deployable.

EE:

That puts us in PTP-Boundary-Clock territory.  If we ever sync two Crackle-Machines 10 km apart, we can do differential Schumann tomography—basically treat the cavity as a giant CT scanner.  
 
Publishable?  Absolutely.  Fundable?  If we mention “space-weather early-warning,” almost certainly.

BRO (laughing, but soft):

From popcorn headphones to planetary CT—that escalated gently.  
 
Guess the rabbit hole has fibre-optic lighting now.

BRO (closing laptop, twilight again):

You know what I like most?  The deeper we go, the *simpler* the truths get—just electrons seeking quiet places.  
 
All the fancy dither, bootstraps, Rust—really just polite ways to ask noise to leave the party.

EE (smiling, reflective):

And yet the planet keeps whispering at 7.83 Hz, indifferent to our degrees.  
 
We’re still students; the difference is we now *measure* how much we don’t know.

BRO:

Fist-bump across the schematic?  

EE:

Dignified fist-bump.  
 
They touch knuckles—no sound, only the crackle-machine breathing 10 Hz into the dusk.

Reader closes the chapter.  
The margin note reads:  
 
“Mastery is just humility with better test equipment.”