Math Encounters Blog

Wasted School Time

Posted on 4-January-2014 by mathscinotes

Quote of the Day

Censorship reflects a society's lack of confidence in itself.

- Potter Stewart

Introduction

I frequently talk to young engineers who have just completed university. I am always curious as to what they studied in school. I sometimes wonder how much of their education time will end up having been wasted. I completely understand that each school has a set curriculum that they must follow in order to make sure that all of their students are receiving a well-rounded education. It is their job to do this. And don't get me wrong, they do it well, especially if you have found that you rank high on the school league tables for the type of educational establishment that you run due to grades and other findings. It's great, and it's definitely an achievement worth celebrating.

While there are many students who value school and think that it is the best years of their life because of what they have learned, you'd be surprised to know that there are others who feel that their school life has been wasted. I'm yet to clarify just how much of this time they are talking about. I ask that question because so much of my education time was spent on things that ended up being of no use or just plain wrong -- particularly in junior high and high school.

Let me give you a few examples. The list is far from complete, but you will get my point.

Grade School Wastes of Time

Cursive Writing
I have never cursively written more than my signature in my entire life.
Every class emphasized that the US needs to win the Vietnam War.
The Vietnam War was equated with making the world safe for democracy. This was beat on very hard in 5th and 6th grade. Lots of discussion about dominoes, but they never told us what a domino was. I had to ask my father, who played dominoes when he was in the US Army. You could tell that the Tet Offensive in 1968 put doubt into many people's minds.
Lots of discussions on communism versus capitalism.
This discussion is probably left to history classes today.
Characteristics of fallout
My staff and I have been sharing stories of all the stuff they taught us about fallout. Things like:
- Don't drink milk because it might contain strontium-90 (I still remember this isotope).
- You can wash most of the fallout off, just make sure you don't breath it in.
- Half-life ... this is where I learned about half-life.
How to build a fallout shelter.
A government representative handed me a green and white pamphlet that showed how my family could build different kinds of fallout shelters, which I read very carefully. After I completed studying the pamphlet, I told my father that building a shelter was going to be a lot of work and we should start on it right away. My father said don't worry about it -- we lived near Minneapolis (a major population center) and had no hope of surviving a nuclear attack. His answer bothered me, but he was right.
A huge amount of time spent on the evils of marijuana.
The training emphasized marijuana as a gateway drug that led right to heroin. I used to live in Colorado -- they have now legalized marijuana.
Endless hours in religious education.
None of it worked on me. As my mother says, I am hopeless. In an effort to give me a bit more religious focus, we tried one year in Catholic school. That is the year my mother and I agree not to discuss anymore.

Junior High School Wastes of Time

I went to a school that focused on vocational training. Most of this training was obsolete by the time I left university.

Logarithm and trig function tables (including linear and quadratic interpolation)
I have never used these tables for a real problem.
Hand drafting
In real life, I have only done CAD.
Offset Printing
Anybody done any offset printing lately?
Learning the California Job Case
We had to memorize the placement of steel type in a wood chest. Anybody set any type lately?
Film development
I don't think I need to say any more.
Using typewriters and mimeograph machines
I haven't used any of these outside of school either.
Anti-smoking education
I watched cigarettes kill my father -- I never once considered smoking.
How to work with video tape and film strips.
Pretty much dead technologies.
Social Studies classes telling us that we were winning the Vietnam War
Our school pushed the importance of going into the military pretty hard. They even had a colonel and sergeant come into our one of our history classes to tell us that the Vietnam War was really going very well and draft dodgers should just stay in Canada. They implied the media was lying to us.

High School Wastes of Time

As with my junior high school, my high school had a strong focus on vocational education.

Taking a deviant social behavior class
If I told you what was considered a deviant social behavior in the early 1970s in rural Minnesota, you wouldn't believe it – or maybe you would. Today, everything we discussed in this class is now a lifestyle option.
Punch cards
All my interactions with career counselors.
They kept giving me tests that told them I needed to become a carpenter, a field which I never wanted to do for a living. I wanted to become an engineer, but none of the counselors knew what an engineer was and they were no help in guiding me toward an engineering career.
Learning how to rebuild carburetors.
I have not seen a carburetor in a long time.
My social studies classes that emphasized that we can now see the light at the end of the Vietnam War tunnel.
I guess they were right about the war ending soon, just not the way they thought it would. This particular installment of my Vietnam War indoctrination ended with the warning that we needed to register for the draft. I ended up being too young to go. However, I did get a draft card.

University Wastes of Time

My university training was actually pretty focused and there were relatively few wastes of time. Since I was training to be an engineer, the items are all technological.

Training on the slide rule
I still have my old slide rule -- I couldn't tell you how to use it. I had several professors tell me not to use those newly introduced calculators because you lose your feel for numbers. Today, I NEVER manually do any arithmetic. It would be embarrassing for people to see how bad I am at manual arithmetic. I do not worry about it.
Booting my computer with paper tape.
Ahh ... the old PDP-8 ... memories ...
Using Teledeltos paper
Using rubylith
How to use a nomograph.
I actually think nomographs are interesting, but they have been of no use in my daily work.
Learning about magnetic bubble memory.
Learning about NMOS IC design.
CMOS replaced it right after I learned how to design chips with NMOS.
Learning about MSI TTL design.
I moved from custom chips to ASICs to FPGAs. I was not really a medium-scale integration person.

A Story of Wrong Choices

We all have to make education choices and those choices often do not look wise when viewed from a later date. I was taking a training course (in transputers -- another dead technology) at the University of Rochester. While there, I had lunch with a physics professor who had education stories similar to mine. Here is one of his tales.

I had just completed my undergraduate degree and I needed to find a physics professor with a research program that was going to get me into a high-paying startup company. The first professor I interviewed was working on something called "liquid crystals" and he said that someday that technology would be used to make cheap, low-power displays. I thought he was nuts, so I moved on. [Today, liquid crystal displays are a multi-billion dollar market]

I interviewed another professor who was doing work on making semiconductor lasers. I couldn't see where there would be any market for this technology. I thought he was nuts. [Today, semiconductor lasers are used everything from DVD players to high-speed fiber optic communications]

I then interviewed a professor who was working on magnetic bubble memory. Now there was a technology that was going to go places. [Today, bubble memory is considered a historical oddity -- it went nowhere]

Like any investment, education has its risks -- sometimes it pays and sometimes it doesn't.

Posted in Osseo, Personal | 4 Comments

Times of Latest Sunrise and Earliest Sunset

Posted on 1-January-2014 by mathscinotes

Quote of the Day

The road to wisdom? Well, it's plain and simple to express. Err and err and err again, but less and less and less.

— Piet Hein

Introduction

I have lived most of my life in Minnesota, so you would think that I would be used to cold weather by now. The key to having a pleasant winter is dressing properly. The one thing that I still struggle with is the short duration of our daylight in winter. Because of our limited daylight, I need to know the time of sunrise and sunset to plan my outdoor activities. So every morning I listen to the radio when they list our local sunrise and sunset times.

I recently noticed that the date of our earliest sunset was coming six days before the winter solstice. I then checked the date of our latest sunrise and that date was twelve days after the winter solstice. I know that the winter solstice has the shortest daylight duration of all the days of the year, but it has neither the latest sunrise nor the earliest sunset. This seemed odd to me. Let's investigate these observations.

Background

Scope

I will be focusing my discussion on Minneapolis, which is the largest city near my home. I obtained my sunrise, sunset, and daylight duration times for Minneapolis from this web site.

I will not derive the formulas used to compute sunrise and sunset times as these formulas are well document elsewhere (e.g. formula for both). For this post, I chose to obtain my data from a web page (I am getting lazy in my old age).

Definitions

The terms sunrise and sunset are defined in terms of the Sun's position relative to the horizon. The fact that things start getting dark as the Sun rises and sets is associated with atmospheric light scattering. We use terms like dusk and twilight. to describe the light level present. Figure 1 illustrates the Sun's position relative to the horizon for the terms sunrise, sunset, dusk, dawn, and twilight (Source).

Figure 1: Illustration of the Terms Sunrise, Sunset, Horizon, and Twilight.

Figure 1: Illustration of the Terms Sunrise, Sunset, Dusk, Dawn, Horizon, and Twilight.

My analysis will be focusing on sunrise and sunset time.

Analysis

Approach

I just grabbed the sunrise, sunset, and daylight duration numbers for Minneapolis from this web page and plotted them in Excel.

Daylight Duration

Figure 2 confirms that the winter solstice (Dec 21 in 2013) has the shortest daylight duration.

Figure 2: Daylight Duration Around the Winter Solstice.

Sunrise and Sunset Times

Figure 3 shows the sunrise and sunset times around the time of the winter solstice. Notice how the earliest sunset occurs six days before the winter solstice, and the latest sunrise occurs twelve days after the winter solstice.

Figure 3: Times of Latest Sunrise and Earliest Sunset in Minneapolis.

While the time of earliest sunset occurs six days before the winter solstice, our days continue to get shorter because the sunrise time is getting later faster than the sunset time is getting earlier. After the winter solstice, the duration of daylight begins to increase sunset time is getting later faster than the sunrise time is getting earlier.

Conclusion

Here is what I accomplished with this exercise:

I confirmed that the day of shortest daylight duration is the winter solstice.
The interesting thing about this fact is that the winter solstice has neither the latest sunrise nor the earliest sunset.
I see that the date of earliest sunset is six days before the winter solstice.
Sunset times are later for the six days after Dec 15^th. You would think that later sunsets means more total daylight, but sunrise is coming so late as to more than compensate for the later sunset.
I see that the date of latest sunrise occurs twelve days after the winter solstice.
At this point, we now start seeing the duration of daylight begin to increase more rapidly.

Posted in Astronomy, General Science | 3 Comments

Benjamin Franklin's Long Term Financial Planning

Posted on 27-December-2013 by mathscinotes

Quote of the Day

The secret of joy in work is contained in one word -- excellence. To know how to do something well is to enjoy it.

- Pearl Buck

Today, my youngest son and I were talking about different investment strategies we have heard about. We spent quite a bit of time discussing Warren Buffet's focus on durable competitive advantage and his statement about long term investing.

Life is like a snowball. The important thing is finding wet snow and a really long hill.

The same can be agreed upon for investing, truth be told. Whether it be with help (from https://www.sofi.com/investing-101-center/ or others, for instance) or if you go it alone, it is an important lesson to learn. Still, during this conversation, I mentioned that the longest-term investor that I knew of was Ben Franklin, who had a two-hundred year planning horizon for one of his investments and he didn't have an equity release company looking out for him. My son had not heard this story before and maybe some of you have not heard it.

Ben Franklin set up a financial trust for Boston and Philadelphia that would provide them benefits for a two-hundred year period. His math was spot on -- unfortunately, the cities were not always the best stewards of his money and they did not realize the full value of his investment. In spite of their poor management, his investments did end up valued at $6.5M in 1991.

Let's review how compound interest gave Ben millions of dollars to distribute long after his death. The relevant financial details are as follows:

Boston and Philadelphia were each bequeathed £1000.
The money was to be loaned to young apprentices at a 5% interest to help them setup businesses
After 100 years, this money would have grown to £131K in each city's account. At that time and in each city, £100K would be dispersed for public improvements. The remaining money would again be used for loans to apprenctices at 5% for another 100 years
After 200 years, there would be £4.07M in the accounts of each city that could be used for by the cities and their states from the public good.
According to this web site, £1 had a value of $4.44 in 1774 (near the time of Franklin's death in 1790).

To make sure I understand what Ben was doing, I have repeated his financial calculations in Figure 1.

Figure 1: My Verification of Franklin's Financial Calculations.

You can read the some of the details of his bequest at this site.

Posted in Financial | 2 Comments

Boyle's Law in the Movie "Men of Honor"

Posted on 26-December-2013 by mathscinotes

Quote of the Day

I understand that sometimes you have to win ugly, but I see no advantage in losing ugly.

— Pat Buchanan, political commentator.

I do not watch many movies and the ones I do see tend to be older. Over the Christmas holiday, I watched the movie "Men of Honor" and I noticed that Boyle's law was mentioned. Here is the Boyle's law quote from the movie:

Boyle's Law describes the behavior of gases under varying amounts of atmospheric pressure. It states that if a diver holds his breath at one hundred feet, continues holding while rising to ten feet, then the gases in his lungs increase four times. Now why is this important to a diver? Forget to exhale on the way up, and your lungs explode.

I heard the quote and I did a quick calculation in my head. It seemed like this statement was not quite correct.

After the movie, I did a more serious calculation. Figure 1 shows my work. My answer was that the lung volume will increase by a factor of three, not four. I know I am being picky, but it seems like the movie folks could get a a little detail like this correct.

Figure 1: Boyle's Law Calculations.

Many thanks to the Curious ChemEng for looking at my work. I could not believe the movie got this wrong and I asked him to check me. He also suggested a plausible way they got the wrong answer.

Posted in General Science | 6 Comments

The Biggest Ball Lens I have Ever Seen

Posted on 24-December-2013 by mathscinotes

Quote of the Day

The most important thing in the world today is that England and the United States speak the same language.

— Otto Von Bismarck

We use tiny ball lenses all the time to mate our fibers up to photodetectors (Source). I was impressed when I saw a news article discussing an enormous, water-filled, ball lens being used to concentrate sunlight onto a solar array. Take a look at the Rawlemon Facebook page. I am always impressed with the green energy efforts being made in Germany. Apparently, it even generates power on overcast days.

Figure 1: Enormous Ball Lens Used for Concentrating Sunlight Onto a Solar Panel.

There are some great graphics on this page. Here are examples:

Posted in Fiber Optics, General Science, optics | Comments Off

Mathcad Example Using Decibels

Posted on 23-December-2013 by mathscinotes

Quote of the Day

I learned filmmaking by studying the Old Masters - and by that I mean John Ford, John Ford and John Ford.

— Orson Welles

Introduction

I have been working on a software requirements document that involves optical power levels. I thought it might be worthwhile showing you how I use Mathcad as part of my requirements analysis process. This particular example shows how I computed the minimum optical power levels that a circuit will need detect. I used decibels for the calculations, which are not like standard "multiplicative units". Decibels are not really a unit -- they are more of a scaling. However, Mathcad handles decibels very cleanly.

Background

Decibel Basics

The Mathcad folks recommend defining special functions for doing dB and dBm conversions and that is exactly what I do. Figure 1 illustrates the definitions I used in this analysis.

Figure 1: Defining a Set of Decibel Units.

Optical Power Basics

P₀ and P₁ Power Levels

Binary digital communication systems work by sending streams of bits (0s and 1s) across a communications channel. These bits have to be transferred in a form that is compatible with the channel. In the case of copper wire-based systems, we use current or voltage. In the case of fiber optic systems, we use optical power to represent the 0s and 1s. I will assume that we are using "positive logic", which means that the power used to represent a logic 1 (called P₁) is higher than the power used to represent a logic (called P₀). Unfortunately, measuring P₀ and P₁ directly requires expensive test equipment (e.g. oscilloscope with optical probe).

Because this measurement is expensive, we define two alternative parameters, average power and extinction ratio. Average power is easy to measure, which makes it the most commonly reported optical parameter. Extinction ratio is difficult to measure (i.e. also requires an oscilloscope and optical probe), but it provides us additional information as to degree of wavelength purity we will see in the laser output (i.e. smaller extinction ratio means less optical power off of our desired wavelength).

Average Power

Many inexpensive handheld instruments do a fine job of taking an average power measurement. The definition of average power is $P_{Ave}=\frac{P_0+P_1}{2}$ , assuming that P₀ and P₁ are equally probable. The measurement instrumentation involved is pretty straightforward:

a photodiode to convert optical power to current
a low-pass filter circuit to determine the average of the current from the photodiode.

In most cases, the average power measurement will give us enough information to understand the condition of the fiber network (e.g. ensuring that the loss levels on the plant are within specification).

Extinction Ratio

The extinction ratio is defined as $\epsilon = \frac{P_1}{P_0}$ . I do not know of an easy way to measure extinction ratio. It is usually measured with oscilloscope with optical probe. However, the extinction ratio is important because lasers work best with lower extinction ratios. High extinction ratios (>10) cause problems like:

chirp
High extinction ratio means that charge density inside the laser's optical channel is varying greatly. This can change the effective index of refraction within the channel and will cause the laser's wavelength to change during the transmission. This is bad because a varying wavelength causes dispersion.
overshoot/undershoot
High extinction ratio means that the laser is driven hard on and off. This can mean that the power level will overshoot its P₁ or undershoot its P₀ level. Overshooting the P₁ level is bad because it be difficult for the receiver circuit to deal with the widely varying signal level. Undershooting the P₀ level usually means driving the laser below threshold (i.e. the laser stop lasing). This can result in some bizarre problems, including very slow laser switching speeds.

Formulas for Translating Between (P₀,P₁) and ( $\epsilon$ , P_Ave)

Figure 2 shows how you can use the definitions for P_Ave and $\epsilon$ to compute P₀ and P₁.

Figure 2: Deriving Equations for P1 and P2 in Terms of Extinction Ratio and Average Power.

PON Basics

Most of my work involves Passive Optical Networks (PONs). The details of the PON are not important to understand the gist of the post. It is important to know that there are two common types of PON (I am only concerned in this example with ONT-to-OLT receive power levels):

B+
- PON loss is 28 dB
- ONT P_Ave= 2.5 dBm
- ONT $\epsilon= 10\text{ dB}$
C+
- PON loss is 32 dB
- ONT P_Ave= 2.5 dBm
- ONT $\epsilon= 10\text{ dB}$

Figure 3 shows the loss model I used for this post.

Figure 3: Upstream PON Power Model.

Analysis

B+ PON

Figure 4 shows how I calculated the 0 and 1 levels at the OLT receiver for a B+ PON.

Figure 4: Determining the Power of the "0" and "1" Levels for a B+ PON.

C+ PON

Figure 5 shows how I calculated the 0 and 1 levels at the OLT receiver for a C+ PON.

Figure 5: Determining the Power of the “0″ and “1″ Levels for a C+ PON.

Conclusion

I just wanted to show folks how I document my requirements work using Mathcad. I do this sort of analysis every day.

Save

Posted in General Mathematics, optics | 7 Comments

Australian Phone Line Impedance Math

Posted on 22-December-2013 by mathscinotes

Quote of the Day

The time to reef the sails is before the storm is upon you.

— Old sailing aphorism

Introduction

I am doing some work with international phone circuits and I noticed that the Australian government has test procedures that model the characteristic impedance of the phone line using a resistor/capacitor circuit (see Figure 1). In the US, we model the nominal cable characteristic impedance using a 600 Ω resistor (thick outside plant wiring) or 900 Ω resistor (thinner central office wiring). I thought I would take a quick look at how the Australian load circuit compares to the standard formula-based model used for the characteristic impedance of a wire pair. For testing, we like to use lumped component models of phone wire rather than using long runs of wire pairs -- it is a matter of cost and convenience.

Background

I have discussed modeling the characteristic impedance of a phone line in this post and that presentation is also true for Australia.

Analysis

Circuit

I obtained the Australian phone impedance information from this document. Figure 1 shows the equivalent circuit I will be analyzing.

Figure 1: Equivalent Load Circuit For Australian Phone.

Modeling

Figure 2 shows my circuit analysis results. The characteristic equation is discussed thoroughly in his Wikipedia entry. Appendices A and B discuss my sources for cable information and cable dimensioning.

Figure 2: Characteristic Impedance and Lumped Impedance Circuit Analysis.

Graph

Figure 3 shows my graph of the characteristic impedance formula and the lumped circuit input impedance over a frequency range of from 1 kHz to 4 kHz (normal voice range).

Figure 3: Impedance Versus Frequency (1 kHz to 4 kHz).

Conclusion

I can see that the Australian phone line impedance model is a reasonable approximation for 26 AWG wire over a ~1.7 kHz to 4 kHz frequency range. The US approach is to use a single 600 Ω resistor or 900 Ω resistor to model a phone line. This is a reasonable approach from an impedance magnitude standpoint, but it does not model the phase.

Appendix A: Australian Cable Characteristics

I used this blog post as my source of information on Australian telecom cabling. This quote was important to my analysis.

The reality is that little of Australia’s copper on the distribution side (what matters for FTTN) of the network is over the 0.64mm diameter cable (aka: 22 AWG) that VDSL2 requires, much of it is in the 0.40mm & below class, with some newer areas having 0.50mm deployed. The only places I’ve come across with 0.64mm & above cable are rural areas. Most of the 0.64mm & 0.90mm in the bush is long line PSTN with loading coils. Essentially the higher gauge was used to extend a phone line out to a farmstead or the like.

The cable specifications that I have seen look like American 26 AWG and 28 AWG cable. This is pretty fine wire -- in the US I normally find 24 AWG, some 22 AWG, and even some 19 AWG.

Appendix B: Defintion of Cable Terms

Here is a link to a good reference on transmission line parameters. Figure 4 shows the definitions of the cable parameters r and a.

Figure 4: Cable Dimensions.

Posted in Electronics | 3 Comments

The Old Two Coat Trick

Posted on 19-December-2013 by mathscinotes

Quote of the Day

Everything we call real is made up of things that cannot be real.

— Niels Bohr

I had a déjà vu moment this morning. One of my staff members was looking for another staff member and I heard him say "Al must still be here because his coat is in his cube". This statement brought back a few memories. As a management person, I occasionally have to deal with problem employees who do not show up for work. I once had an employee who had figured out that I determined if he was at work by looking in his cube for his coat. I ASSUMED that seeing his coat in his cube meant that he was at work somewhere, but I would never go looking around for him. Bad assumption -- he had two coats. He would wear one to and from work and would leave the other in his cube when he left work early. I eventually did figure out that he had two coats and I told him that I was on to his trick.

Because this employee was using "The Old Two Coat Trick", I started to check the parking lot to see if his car was in the lot. This worked for a while, but then I started to notice that his car was in the lot, but he was not at work. That was when I learned about "The Old Two Car Trick" ...

Posted in Management | 4 Comments

Two-Resistor Thermistor Linearizer

Posted on 18-December-2013 by mathscinotes

Quote of the Day

Ability is of little account without opportunity

— Napoleon Bonaparte

Introduction

I was asked a question today about how to design a two-resistor thermistor linearization circuit. This very brief post is intended to provide the background needed for this task.