## Wye Delta Tee Pi: Observations on Three-Terminal Networks

Today I’m going to talk a little bit about three-terminal linear passive networks. These generally come in two flavors, wye and delta.

Why Wye?The town of Why, Arizona has a strange name that comes from the shape of the original road junction of Arizona State Highways 85 and 86, which was shaped like the letter Y. This is no longer the case, because the state highway department reconfigured the intersection

## The Least Interesting Circuit in the World

It does nothing, most of the time.

It cannot compute pi. It won’t oscillate. It doesn’t light up.

Often it makes other circuits stop working.

It is… the least interesting circuit in the world.

What is it?

About 25 years ago, I took a digital computer architecture course, and we were each given use of an ugly briefcase containing a bunch of solderless breadboards and a power supply and switches and LEDs — and a bunch of

## Linear Feedback Shift Registers for the Uninitiated, Part XVIII: Primitive Polynomial Generation

Last time we figured out how to reverse-engineer parameters of an unknown CRC computation by providing sample inputs and analyzing the corresponding outputs. One of the things we discovered was that the polynomial \( x^{16} + x^{12} + x^5 + 1 \) used in the 16-bit X.25 CRC is not primitive — which just means that all the nonzero elements in the corresponding quotient ring can’t be generated by powers of \( x \), and therefore the corresponding 16-bit LFSR with taps in bits 0, 5,...

## R1C1R2C2: The Two-Pole Passive RC Filter

I keep running into this circuit every year or two, and need to do the same old calculations, which are kind of tiring. So I figured I’d just write up an article and then I can look it up the next time.

This is a two-pole passive RC filter. Doesn’t work as well as an LC filter or an active filter, but it is cheap. We’re going to find out a couple of things about its transfer function.

First let’s find out the transfer function of this circuit:

Not very...

## Linear Feedback Shift Registers for the Uninitiated, Part XVII: Reverse-Engineering the CRC

Last time, we continued a discussion about error detection and correction by covering Reed-Solomon encoding. I was going to move on to another topic, but then there was this post on Reddit asking how to determine unknown CRC parameters:

I am seeking to reverse engineer an 8-bit CRC. I don’t know the generator code that’s used, but can lay my hands on any number of output sequences given an input sequence.

This is something I call the “unknown oracle”...

## Linear Feedback Shift Registers for the Uninitiated, Part XVI: Reed-Solomon Error Correction

Last time, we talked about error correction and detection, covering some basics like Hamming distance, CRCs, and Hamming codes. If you are new to this topic, I would strongly suggest going back to read that article before this one.

This time we are going to cover Reed-Solomon codes. (I had meant to cover this topic in Part XV, but the article was getting to be too long, so I’ve split it roughly in half.) These are one of the workhorses of error-correction, and they are used in...

## Linear Feedback Shift Registers for the Uninitiated, Part XV: Error Detection and Correction

Last time, we talked about Gold codes, a specially-constructed set of pseudorandom bit sequences (PRBS) with low mutual cross-correlation, which are used in many spread-spectrum communications systems, including the Global Positioning System.

This time we are wading into the field of error detection and correction, in particular CRCs and Hamming codes.

Ernie, You Have a Banana in Your EarI have had a really really tough time writing this article. I like the...

## Linear Regression with Evenly-Spaced Abscissae

What a boring title. I wish I could come up with something snazzier. One word I learned today is studentization, which is just the normalization of errors in a curve-fitting exercise by the sample standard deviation (e.g. point \( x_i \) is \( 0.3\hat{\sigma} \) from the best-fit linear curve, so \( \frac{x_i - \hat{x}_i}{\hat{\sigma}} = 0.3 \)) — Studentize me! would have been nice, but I couldn’t work it into the topic for today. Oh well.

I needed a little break from...

## Linear Feedback Shift Registers for the Uninitiated, Part XIV: Gold Codes

Last time we looked at some techniques using LFSR output for system identification, making use of the peculiar autocorrelation properties of pseudorandom bit sequences (PRBS) derived from an LFSR.

This time we’re going to jump back to the field of communications, to look at an invention called Gold codes and why a single maximum-length PRBS isn’t enough to save the world using spread-spectrum technology. We have to cover two little side discussions before we can get into Gold...

## Linear Feedback Shift Registers for the Uninitiated, Part XIII: System Identification

Last time we looked at spread-spectrum techniques using the output bit sequence of an LFSR as a pseudorandom bit sequence (PRBS). The main benefit we explored was increasing signal-to-noise ratio (SNR) relative to other disturbance signals in a communication system.

This time we’re going to use a PRBS from LFSR output to do something completely different: system identification. We’ll show two different methods of active system identification, one using sine waves and the other...

## Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields

Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.

— Évariste Galois, May 29, 1832

I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical understanding, with some practical...

## Byte and Switch (Part 1)

Imagine for a minute you have an electromagnet, and a microcontroller, and you want to use the microcontroller to turn the electromagnet on and off. Sounds pretty typical, right?We ask this question on our interviews of entry-level electrical engineers: what do you put between the microcontroller and the electromagnet?We used to think this kind of question was too easy, but there are a surprising number of subtleties here (and maybe a surprising number of job candidates that were missing...

## How to Estimate Encoder Velocity Without Making Stupid Mistakes: Part II (Tracking Loops and PLLs)

Yeeehah! Finally we're ready to tackle some more clever ways to figure out the velocity of a position encoder. In part I, we looked at the basics of velocity estimation. Then in my last article, I talked a little about what's necessary to evaluate different kinds of algorithms. Now it's time to start describing them. We'll cover tracking loops and phase-locked loops in this article, and Luenberger observers in part III.

But first we need a moderately simple, but interesting, example...

## Ten Little Algorithms, Part 3: Welford's Method (and Friends)

Other articles in this series:

- Part 1: Russian Peasant Multiplication
- Part 2: The Single-Pole Low-Pass Filter
- Part 4: Topological Sort
- Part 5: Quadratic Extremum Interpolation and Chandrupatla's Method
- Part 6: Green’s Theorem and Swept-Area Detection

Last time we talked about a low-pass filter, and we saw that a one-line...

## Thermistor signal conditioning: Dos and Don'ts, Tips and Tricks

In an earlier blog entry, I mentioned this circuit for thermistor signal conditioning:

It is worth a little more explanation on thermistor signal conditioning; it's something that's often done poorly, whereas it's among the easiest applications for signal conditioning.

The basic premise here is that there are two resistors in a voltage divider: Rth is the thermistor, and Rref is a reference resistor. Here Rref is either R3 alone, or R3 || R4, depending on the gain...

## Lost Secrets of the H-Bridge, Part I: Ripple Current in Inductive Loads

So you think you know about H-bridges? They're something I mentioned in my last post about signal processing with Python.

Here we have a typical H-bridge with an inductive load. (Mmmmm ahhh! It's good to draw by hand every once in a while!) There are four power switches: QAH and QAL connecting node A to the DC link, and QBH and QBL connecting node B to the DC link. The load is connected between nodes A and B, and here is represented by an inductive load in series with something else. We...

## Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine

Today’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.

One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to avoid overflow. Adding or subtracting two 16-bit integers produces a 17-bit result; multiplying two 16-bit integers produces a 32-bit result. In fixed-point arithmetic we typically multiply and shift right; for example, if we wanted to multiply some...

## Second-Order Systems, Part I: Boing!!

I’ve already written about the unexciting (but useful) 1st-order system, and about slew-rate limiting. So now it’s time to cover second-order systems.

The most common second-order systems are RLC circuits and spring-mass-damper systems.

Spring-mass-damper systems are fairly common; you’ve seen these before, whether you realize it or not. One household example of these is the spring doorstop (BOING!!):

(For what it’s worth: the spring...

## My Love-Hate Relationship with Stack Overflow: Arthur S., Arthur T., and the Soup Nazi

Warning: In the interest of maintaining a coherent stream of consciousness, I’m lowering the setting on my profanity filter for this post. Just wanted to let you know ahead of time.

I’ve been a user of Stack Overflow since December of 2008. And I say “user” both in the software sense, and in the drug-addict sense. I’m Jason S, user #44330, and I’m a programming addict. (Hi, Jason S.) The Gravatar, in case you were wondering, is a screen...

## Slew Rate Limiters: Nonlinear and Proud of It!

I first learned about slew rate limits when I was in college. Usually the subject comes up when talking about the nonideal behavior of op-amps. In order for the op-amp output to swing up and down quickly, it has to charge up an internal capacitor with a transistor circuit that’s limited in its current capability. So the slew rate limit \( \frac{dV}{dt} = \frac{I_{\rm max}}{C} \). And as long as the amplitude and frequency aren’t too high, you won’t notice it. But try to...

## Lost Secrets of the H-Bridge, Part IV: DC Link Decoupling and Why Electrolytic Capacitors Are Not Enough

Those of you who read my earlier articles about H-bridges, and followed them closely, have noticed there's some unfinished business. Well, here it is. Just so you know, I've been nervous about writing the fourth (and hopefully final) part of this series for a while. Fourth installments after a hiatus can bring bad vibes. I mean, look what it did to George Lucas: now we have Star Wars Episode I: The Phantom Menace and

## How to Build a Fixed-Point PI Controller That Just Works: Part II

In Part I we talked about some of the issues around discrete-time proportional-integral (PI) controllers:

- various forms and whether to use the canonical form for z-transforms (don't do it!)
- order of operation in the integral term: whether to scale and then integrate (my recommendation), or integrate and then scale.
- saturation and anti-windup

In this part we'll talk about the issues surrounding fixed-point implementations of PI controllers. First let's recap the conceptual structure...

## Supply Chain Games: What Have We Learned From the Great Semiconductor Shortage of 2021? (Part 2)

Welcome back! Today we’re going to zoom around again in some odd directions, and give a roundabout introduction to the semiconductor industry, touching on some of the following questions:

- How do semiconductors get designed and manufactured?
- What is the business of semiconductor manufacturing like?
- What are the different types of semiconductors, and how does that affect the business model of these manufacturers?
- How has the semiconductor industry evolved over...

## Round Round Get Around: Why Fixed-Point Right-Shifts Are Just Fine

Today’s topic is rounding in embedded systems, or more specifically, why you don’t need to worry about it in many cases.

One of the issues faced in computer arithmetic is that exact arithmetic requires an ever-increasing bit length to avoid overflow. Adding or subtracting two 16-bit integers produces a 17-bit result; multiplying two 16-bit integers produces a 32-bit result. In fixed-point arithmetic we typically multiply and shift right; for example, if we wanted to multiply some...

## Which MOSFET topology?

A recent electronics.StackExchange question brings up a good topic for discussion. Let's say you have a power supply and a 2-wire load you want to be able to switch on and off from the power supply using a MOSFET. How do you choose which circuit topology to choose? You basically have four options, shown below:

From left to right, these are:

High-side switch, N-channel MOSFET High-side switch, P-channel MOSFET Low-side switch, N-channel...## Development of the MOS Technology 6502: A Historical Perspective

One ubiquitous microprocessor of the late 1970s and 1980s was the MOS Technology MCS 6502. I included a section on the development of the 6502 in Part 2 of Supply Chain Games, and have posted it as an excerpt here, as I believe it is deserving in its own right.

(Note: MOS Technology is pronounced with the individual letters M-O-S “em oh ess”,[1] not “moss”, and should not be confused with another semiconductor company,

## Important Programming Concepts (Even on Embedded Systems) Part IV: Singletons

Other articles in this series:

- Part I: Idempotence
- Part II: Immutability
- Part III: Volatility
- Part V: State Machines
- Part VI: Abstraction

Today’s topic is the singleton. This article is unique (pun intended) in that unlike the others in this series, I tried to figure out a word to use that would be a positive concept to encourage, as an alternative to singletons, but

## Lost Secrets of the H-Bridge, Part III: Practical Issues of Inductor and Capacitor Ripple Current

We've been analyzing the ripple current in an H-bridge, both in an inductive load and the DC link capacitor. Here's a really quick recap; if you want to get into more details, go back and read part I and part II until you've got equations coming out of your ears. I promise there will be a lot less grungy math in this post. So let's get most of it out of the way:

Switches QAH and QAL are being turned on and off with pulse-width modulation (PWM), to produce an average voltage DaVdc on...

## Linear Feedback Shift Registers for the Uninitiated, Part I: Ex-Pralite Monks and Finite Fields

Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.

— Évariste Galois, May 29, 1832

I was going to call this short series of articles “LFSRs for Dummies”, but thought better of it. What is a linear feedback shift register? If you want the short answer, the Wikipedia article is a decent introduction. But these articles are aimed at those of you who want a little bit deeper mathematical understanding, with some practical...

## Lost Secrets of the H-Bridge, Part I: Ripple Current in Inductive Loads

So you think you know about H-bridges? They're something I mentioned in my last post about signal processing with Python.

Here we have a typical H-bridge with an inductive load. (Mmmmm ahhh! It's good to draw by hand every once in a while!) There are four power switches: QAH and QAL connecting node A to the DC link, and QBH and QBL connecting node B to the DC link. The load is connected between nodes A and B, and here is represented by an inductive load in series with something else. We...