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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.”]