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, Part XV: Error Detection and Correction
CRCs and Hamming codes look a lot less magical when you view them as redundancy with a purpose. Jason Sachs walks from parity bits and checksums into finite-field polynomial arithmetic, then shows how CRCs map cleanly onto LFSRs and how Hamming codes use syndromes to locate single-bit errors. It is a practical tour of error detection and correction, with enough worked examples to make the theory feel usable.
Linear Regression with Evenly-Spaced Abscissae
Jason Sachs cuts through the matrix algebra to show a tiny trick for linear regression when x values are evenly spaced. You can compute the intercept as the mean and the slope as a simple weighted sum with arithmetic weights, using q = 12/(m^3 - m). The post includes Python examples and a compact routine to get least-squares coefficients without matrix solvers.
Linear Feedback Shift Registers for the Uninitiated, Part XIV: Gold Codes
Gold codes solve a practical spread-spectrum problem, sharing one PRBS across many transmitters eventually runs into ugly synchronization and correlation issues. Jason Sachs walks through why shifted copies of a single LFSR sequence are not enough, then shows how preferred pairs of m-sequences create a family of Gold codes with bounded cross-correlation. The post wraps with Python experiments and a UART DSSS demo that decodes multiple overlapping messages cleanly.
Lessons Learned from Embedded Code Reviews (Including Some Surprises)
Jason Sachs recounts a round of motor-controller code reviews and the practical lessons his team learned about quality and tooling. He explains how a simple "ready for review" checklist and automated style checks kept meetings focused on substantive issues, and why choosing the right review tool matters after discovering lost comments in Stash. Read for concrete tips on process, subgit mirroring, vera++, and Upsource.
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.
Beware of Analog Switch Leakage Current
Leakage currents in analog switches can quietly wreck precision reference circuits at elevated temperature. Jason M. Sachs walks through three switch-topology implementations for a switchable 1.25 V reference and shows which topology gives the smallest worst-case output error using real part specs. He explains why op amp input bias is usually negligible and gives practical fixes: lower resistances, better switches, or limiting temperature range.
Thermistor signal conditioning: Dos and Don'ts, Tips and Tricks
Jason Sachs shows how to keep thermistor conditioning simple and accurate for embedded systems. He warns against analog linearization and excessive analog stages, and explains why ratiometric dividers, proper ADC buffering, and using the same reference voltage give better results. The post also covers thermal pitfalls like self-heating and lead conduction, plus practical tips for ADC autocalibration and polynomial temperature conversion.
Important Programming Concepts (Even on Embedded Systems) Part V: State Machines
State machines are not glamorous, but they solve a lot of real embedded problems. Jason Sachs uses a motorized couch example to show how FSMs and Harel statecharts expose corner cases, simplify timing constraints, and make behavior easier to specify and review. The article walks through hand-rolled switches, tabular implementations, the state pattern, libraries like QP and Boost, and tool tradeoffs.
Scorchers, Part 3: Bare-Metal Concurrency With Double-Buffering and the Revolving Fireplace
Jason Sachs presents a practical, low-overhead concurrency pattern for tiny bare-metal systems where an ISR (Speedy) must safely exchange data with a nonreal-time main loop (Poky). He describes the "revolving fireplace", a double-buffering variant that swaps ownership of two shared memory regions, and walks through C examples, atomic/volatile considerations, and testing strategies so you can implement it on RAM-constrained MCUs.
How to Estimate Encoder Velocity Without Making Stupid Mistakes: Part II (Tracking Loops and PLLs)
Jason Sachs explains why simple differentiation of encoder counts often fails and how tracking loops and PLLs give more robust velocity estimates. Using a pendulum thought experiment and Python examples, he shows how a PI-based tracking loop reduces noise and eliminates steady-state ramp error, and why vector PLLs with quadrature mixing avoid cycle slips and atan2 unwrap pitfalls in noisy or analog sensing.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Lazy Properties in Python Using Descriptors
Python descriptors let you outsource attribute lookup, and Jason Sachs walks through a practical use: lazy, cached properties. He presents a LazyProperty descriptor that defaults to a WeakKeyDictionary cache so computed results are stored on first access and automatically purged when objects are garbage collected. The post shows how to share caches by value using attrkey or swap cache classes for different use cases.
Slew Rate Limiters: Nonlinear and Proud of It!
Slew-rate limits are a small nonlinear detail that often decides whether a controller behaves nicely or wrecks hardware. Jason Sachs walks through why slew limits appear in electronics and actuators, then shows two practical digital ways to impose limits: constraining input increments and constraining input around the output. He compares performance on underdamped second-order systems, gives closed-form intuition for overshoot, and demonstrates simulations with scipy and ODE solvers.
