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.
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.
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.
Ten Little Algorithms, Part 7: Continued Fraction Approximation
In this article we explore the use of continued fractions to approximate any particular real number, with practical applications.
How to Analyze a Three-Op-Amp Instrumentation Amplifier
The three-op-amp instrumentation amplifier gives you high input impedance, improved net bandwidth, and much lower sensitivity to resistor mismatch than a single-op-amp differential stage. Jason M. Sachs walks through the algebra, numeric examples, and historical notes to show how the preamp isolates common-mode, why splitting gain boosts bandwidth, how overall gain can be set with one resistor, and what practical limits to watch.
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.
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.
Ten Little Algorithms, Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
Today we will be drifting back into the topic of numerical methods, and look at an algorithm that takes in a series of discretely-sampled data points, and estimates the maximum value of the waveform they were sampled from.
Linear Feedback Shift Registers for the Uninitiated, Part VI: Sing Along with the Berlekamp-Massey Algorithm
Jason Sachs breaks down the Berlekamp-Massey algorithm and shows how to recover an LFSR's minimal connection polynomial from a stream of output bits. The article mixes intuition, worked examples, and Python code to demonstrate the update rule, visual debugging tables, and when the solution is unique. Expect practical implementation notes, a complexity discussion, and a libgf2 example you can run in an IPython notebook.
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.
Linear Feedback Shift Registers for the Uninitiated, Part V: Difficult Discrete Logarithms and Pollard's Kangaroo Method
Most discrete-log problems are hopeless by brute force, but clever algorithms cut that cost to feasible levels. This installment walks through baby-step giant-step, Pollard’s rho and kangaroo methods, and how Silver-Pohlig-Hellman and index calculus leverage group structure to speed attacks on GF(2^n) fields. Jason Sachs includes Python examples, heuristics, and complexity nuggets so you can see when each method is practical.
Turn It On Again: Modeling Power MOSFET Turn-On Dependence on Source Inductance
This is a short article explaining how to analyze part of the behavior of a power MOSFET during turn-on, and how it is influenced by the parasitic inductance at the source terminal. The brief qualitative reason that source inductance is undesirable is that it uses up voltage when current starts increasing during turn-on (remember, V = L dI/dt), voltage that would otherwise be available to turn the transistor on faster. But I want to show a quantitative approximation to understand the impact of additional source inductance, and I want to compare it to the effects of extra inductance at the gate or drain.
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.
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 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.
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.
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.
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.
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 VIII: Matrix Methods and State Recovery
Matrix methods for LFSRs look intimidating, but Jason Sachs walks through companion-matrix representations and shows why they matter for time shifts and state recovery. He derives lookahead masks from powers of the companion matrix, then translates those matrix insights into efficient bitwise and finite-field algorithms. The article includes two simple state-recovery methods and working Python/libgf2 examples you can run and adapt.
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.
