Scorchers, Part 2: Unknown Bugs and Popcorn
Jason Sachs likens bug hunting to popping popcorn to explain diminishing returns when preparing a release. He argues that the rate of new bug reports is a practical signal for whether to keep testing or ship, and that late fixes incur hidden costs like extra testing, branching, documentation, and lost focus. The piece also warns that embedded firmware needs stricter pre-release testing because updates are rarer.
Racing to Sleep
Jason Sachs walks through a realistic field sensor case study, the BigBrotherBear 2000, to show how a careful power budget exposes surprising energy costs. He demonstrates that radios and data transmission often dwarf quiescent MCU current, explains the race-to-sleep principle for computation-bound tasks, and outlines practical wake-up and measurement trade-offs so engineers can extend battery lifetime in real deployments.
Jaywalking Around the Compiler
Messing with inline assembly can feel powerful until the compiler silently undoes you. Jason Sachs walks through a real bug on a Microchip dsPIC33E where pushing CORCON and writing a fixed value corrupts compiler-managed state and produces wrong results when the compiler reorders code. The post shows why the stack and certain registers are off-limits to raw inline asm, and gives practical, safe patterns to save and restore mode bits.
Shibboleths: The Perils of Voiceless Sibilant Fricatives, Idiot Lights, and Other Binary-Outcome Tests
Binary tests look simple until you try to pick a threshold, because false positives, false negatives, and base rate all collide. Jason Sachs uses a deliberately absurd detective story, then walks through the math of expected value, medical screening tradeoffs, idiot lights, and even a triage-style three-way decision. The payoff is a practical way to think about when a pass/fail signal helps, and when raw data or a second test is worth the extra complexity.
Wye Delta Tee Pi: Observations on Three-Terminal Networks
Three-terminal passive networks, wye, delta, tee, and pi, are more interchangeable than many engineers expect. Jason Sachs walks through Kennelly's wye-delta formulas, Z and Y matrix representations for tee and pi two-port networks, and worked examples ranging from balanced to highly skewed impedances. The post highlights practical payoffs, including using topology transforms to substitute hard-to-source capacitors with simpler, precision-friendly parts.
The Least Interesting Circuit in the World
Jason Sachs pulls apart the humble power-on reset and shows why the common RC-and-Schmitt trick is the least interesting but most dangerous circuit in your design. He walks through voltage thresholds, brown-out reset behavior, and how slow or noisy Vdd ramps can let parts start in indeterminate states. Read this for practical rules on choosing supervisors, comparators, and reset pulse timing to ensure reliable embedded startup.
Linear Feedback Shift Registers for the Uninitiated, Part XVIII: Primitive Polynomial Generation
Jason Sachs walks through how to find primitive polynomials for GF(2) LFSRs, moving from naive exhaustive checks to smarter synthetic constructions. The article compares sieve and constructive methods, shows practical optimizations like parity checks and companion-matrix updates, and demonstrates decimation plus Berlekamp-Massey to generate all primitives from one seed; it also teases a novel Falling Coyote Algorithm for additional speedups.
R1C1R2C2: The Two-Pole Passive RC Filter
Jason Sachs walks through the math and simulation for the common two-pole passive RC filter, turning repetitive algebra into a compact reference you can reuse. He derives the closed-form transfer function, extracts the natural frequency and damping ratio, and explains why the topology cannot be underdamped without inductors or active stages. The post finishes with a state-space simulation recipe and practical component guidance.
Linear Feedback Shift Registers for the Uninitiated, Part XVII: Reverse-Engineering the CRC
Jason Sachs shows how to pry CRC parameters out of a black-box oracle and reimplement the checksum yourself. By canceling the affine offsets, probing single-bit basis messages, and treating per-bit outputs as LFSR sequences, you can recover the generator polynomial, bit and byte order, and init/final XOR values. The post includes working Python code, a 4-message shortcut, and real-world tests such as zlib CRC32.
Linear Feedback Shift Registers for the Uninitiated, Part XVI: Reed-Solomon Error Correction
Jason Sachs demystifies Reed-Solomon codes with hands-on examples and pragmatic tips for embedded engineers. The article shows why RS encoding is just polynomial division in GF(2^m), why decoding is mathematically heavier, and how to implement encoders in Python and in C-friendly form using LFSRs and table-driven methods. Read this for working code, generator-polynomial examples, and an embedded-minded view of RS practicalities.
Linear Feedback Shift Registers for the Uninitiated
Jason Sachs assembled an eighteen-part deep dive into linear feedback shift registers, connecting the simple shift-register circuit to finite-field algebra and practical tools. The series walks through primitive polynomials, Berlekamp-Massey state recovery, libgf2-based analysis, discrete-log methods, and real-world uses from PRNGs and Gold codes to Reed-Solomon and CRC reverse-engineering. It’s a single reference for engineers who want both theory and working code.
In Memoriam: Frederick P. Brooks, Jr. and The Mythical Man-Month
Fred Brooks’ The Mythical Man-Month is still a surprisingly sharp guide to software projects, and Jason Sachs shows why it matters even more than its old mainframe setting suggests. He revisits Brooks’ ideas on surgical teams, conceptual integrity, throwaway prototypes, and schedule estimation, then maps them to modern embedded and software engineering realities. The result is a tribute, a book review, and a practical reminder that roles, architecture, and testing still make or break delivery.
R1C1R2C2: The Two-Pole Passive RC Filter
Jason Sachs walks through the math and simulation for the common two-pole passive RC filter, turning repetitive algebra into a compact reference you can reuse. He derives the closed-form transfer function, extracts the natural frequency and damping ratio, and explains why the topology cannot be underdamped without inductors or active stages. The post finishes with a state-space simulation recipe and practical component guidance.
Ten Little Algorithms, Part 8: Miller-Rabin Primality Test (and Living with Uncertainty)
Part 8 of the Ten Little Algorithms series: A look at the Miller-Rabin primality test, along with Pollard's rho algorithm for factoring, and some perspectives on very low levels of uncertainty.
Analog-to-Digital Confusion: Pitfalls of Driving an ADC
Wayne's thermistor board showed one ADC channel changing when another was heated, a classic case of ADC input cross-coupling. The post walks through how multiplexed ADCs, the small sample-and-hold capacitor, source impedance, sampling time, repeated sampling rates, and added charge reservoirs interact to create errors. Learn practical fixes including increasing sample time, sizing external caps, adding op-amp buffers, and using an RC dampener with PCB layout tips.
How to Analyze a Differential Amplifier
Jason Sachs walks through the algebra and intuition behind the classic four-resistor differential amplifier. He derives the exact output equation, isolates error terms from resistor mismatch and op-amp imperfections, and explains why common-mode gain depends on mismatch not on the differential gain. Read this for clear formulas, modal insight into common-mode versus differential-mode, and practical steps to reduce offsets in real designs.
Modeling Gate Drive Diodes
This is a short article about how to analyze the diode in some gate drive circuits when figuring out turn-off characteristics --- specifically, determining the relationship between gate drive current and gate voltage during turn-off of a power transistor.
Adventures in Signal Processing with Python
Jason Sachs shows how PyLab (numpy, scipy, matplotlib) can handle many signal-processing and visualization tasks engineers usually reach for MATLAB to do. He walks through practical examples including PWM ripple, two pole RC filters, and symbolic math with SymPy, and shares real-world installation tips and trade-offs. The post closes with pointers to IPython and pandas to speed interactive analysis and data handling.
Ten Little Algorithms, Part 1: Russian Peasant Multiplication
Jason Sachs revisits a centuries-old multiplication trick and shows why it still matters. He lays out Russian Peasant Multiplication with simple Python code, then reveals how the same shift-and-add pattern maps to GF(2) polynomial arithmetic and to exponentiation by squaring. The post mixes historical context with practical bitwise techniques that are useful for embedded and low-level math work.
Someday We’ll Find It, The Kelvin Connection
Low-ohm measurements will fool your multimeter unless you use Kelvin connections. Jason Sachs walks through four-wire sensing using a current-limited supply and two DMMs, explains thermoelectric and connector-related errors, and shows why schematics and PCB layout must reflect Kelvin sense pads to avoid subtle measurement and circuit problems.
Tolerance Analysis
Jason Sachs walks through practical tolerance analysis by designing a 24V overvoltage detector from the ground up, combining resistor tolerances, temperature coefficients, reference and comparator errors, hysteresis, and dynamic RC behavior. He demonstrates worst-case stacking with real datasheet numbers, shows how solder and mechanical stress affect resistor choice, and sizes filtering so the comparator meets a microsecond-range trip requirement. The article is a hands-on guide full of worked examples and trade-offs for embedded hardware engineers.
Lost Secrets of the H-Bridge, Part I: Ripple Current in Inductive Loads
Jason Sachs digs into what PWM switching actually does to current in an H-bridge with an inductive load, and why that ripple matters for motors and power converters. He derives closed-form ripple formulas, shows how to compute a reference current I_R0 = VDC·T/L, and uses Python and sympy to plot and verify results. Read it for practical rules to halve ripple and raise its frequency.
Linear Feedback Shift Registers for the Uninitiated, Part XVI: Reed-Solomon Error Correction
Jason Sachs demystifies Reed-Solomon codes with hands-on examples and pragmatic tips for embedded engineers. The article shows why RS encoding is just polynomial division in GF(2^m), why decoding is mathematically heavier, and how to implement encoders in Python and in C-friendly form using LFSRs and table-driven methods. Read this for working code, generator-polynomial examples, and an embedded-minded view of RS practicalities.
Lost Secrets of the H-Bridge, Part III: Practical Issues of Inductor and Capacitor Ripple Current
Jason Sachs cuts through the math to show what ripple current actually does to H-bridge hardware. He explains why peak current is the limiting factor for inductors, why capacitor ESR usually dominates DC-link voltage ripple, and how center-aligned PWM and duty selection reduce harmonics and ripple. Read this if you want practical rules of thumb and calculation templates for real power-electronics designs.
Important Programming Concepts (Even on Embedded Systems) Part IV: Singletons
Singletons are convenient but often a modularity killer, especially in embedded firmware. Jason Sachs walks through the many faces of singletons, from static members and globals to hardware registers and user-visible application singletons, and shows practical ways to avoid tight coupling. Read this for concrete embedded examples and pragmatic fixes like passing state explicitly, using interfaces or factories, and isolating unavoidable globals in a HAL.
The Least Interesting Circuit in the World
Jason Sachs pulls apart the humble power-on reset and shows why the common RC-and-Schmitt trick is the least interesting but most dangerous circuit in your design. He walks through voltage thresholds, brown-out reset behavior, and how slow or noisy Vdd ramps can let parts start in indeterminate states. Read this for practical rules on choosing supervisors, comparators, and reset pulse timing to ensure reliable embedded startup.
Second-Order Systems, Part I: Boing!!
Jason Sachs takes the spring 'boing' of a doorstop into the math of second-order systems, using the series LRC circuit as a concrete example. He shows two standard transfer-function forms, explains why ωn only scales time while ζ sets the response shape, and derives pole locations plus an exact overshoot formula that helps tune embedded-system responses.
Two Capacitors Are Better Than One
Jason Sachs revisits a simple stacked RC trick that dramatically reduces DC error from capacitor insulation leakage in long time-constant filters. Splitting one RC into two stages forces most of the DC drop onto the lower capacitor, squaring the remaining error while changing the effective pole locations. The post walks through the math, practical component tradeoffs, and when to prefer a digital approach.
Padé Delay is Okay Today
High-order Padé approximations for time delays break in surprising ways, but the failure is not magic. Jason Sachs walks through why coefficient-based transfer functions and companion-form state-space are numerically fragile, shows how to compute poles and zeros directly from the hypergeometric form with Newton iteration, and demonstrates building modal or block-diagonal state-space realizations to make high-order Padé delays practical while noting remaining limits.
Byte and Switch (Part 2)
Running a thermistor front end from a single AA cell exposes problems you might not expect. Jason Sachs walks through a switchable-gain divider using a P-channel MOSFET and shows how MOSFET off-state leakage and low supply voltages can corrupt high-impedance temperature readings. The post compares bipolar transistors and analog switch ICs as fixes and gives practical component guidance for one-cell designs.
